ReviewArticle Computational Methods for Ab Initio ...

Review ArticleComputational Methods for Ab Initio Molecular Dynamics

Eric Paquet 1 and Herna L Viktor2

1National Research Council 1200 Montreal Road Ottawa ON Canada2University of Ottawa 800 King Edward Avenue Ottawa ON Canada

Correspondence should be addressed to Eric Paquet ericpaquetnrc-cnrcgcca

Received 23 February 2018 Accepted 20 March 2018 Published 29 April 2018

Academic Editor Yusuf Atalay

Copyright copy 2018 Eric Paquet and Herna L Viktor This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Ab initio molecular dynamics is an irreplaceable technique for the realistic simulation of complex molecular systems and processesfrom first principles This paper proposes a comprehensive and self-contained review of ab initio molecular dynamics from acomputational perspective and from first principles Quantum mechanics is presented from a molecular dynamics perspectiveVarious approximations and formulations are proposed including the Ehrenfest BornndashOppenheimer andHartreendashFockmoleculardynamics Subsequently the KohnndashSham formulation ofmolecular dynamics is introduced as well as the afferent concept of densityfunctional As a result CarndashParrinello molecular dynamics is discussed together with its extension to isothermal and isobaricprocesses CarndashParrinello molecular dynamics is then reformulated in terms of path integrals Finally some implementation issuesare analysed namely the pseudopotential the orbital functional basis and hybrid molecular dynamics

1 Introduction

Ab initio molecular dynamics (AIMD) is an irreplaceabletechnique for the realistic simulation of complex molecularsystems and processes associated with biological organisms[1 2] such as monoclonal antibodies as illustrated in Figure 1With ab initio molecular dynamics it is possible to predict insilico phenomena for which an in vivo experiment is eithertoo difficult or expensive or even currently deemed infeasible[3 4] Ab initio molecular dynamics essentially differs frommolecular dynamics (MD) in two ways Firstly AIMD isbased on the quantum Schrodinger equation while its clas-sical counterpart relies on Newtonrsquos equation Secondly MDrelies on semiempirical effective potentials which approxi-mate quantum effects while AIMD is based on the real physi-cal potentials

In this paper we present a review of ab initio moleculardynamics from a computational perspective and from firstprinciples Our main aim is to create a document which isself-contained coherent and concise but still as completeas possible As the theoretical details are essential for real-life implementation we have provided the equations for therelevant physical aspect and approximations presented Most

important we expose how these equations and concepts arerelated to one another

The paper is organised as follows In Section 2 quantummechanics is reviewed from a molecular dynamics perspec-tive Two important approximations are introduced namelythe adiabatic and the BornndashOppenheimer approximationsSubsequently in Section 3 the Ehrenfest molecular dynamicsis presented which allows for a semiclassical treatment ofthe nuclei This opens the door in Section 4 to the BornndashOppenheimer molecular dynamics formulation This is fol-lowed in Section 5 by the HartreendashFock formulation whichapproximates the atomic antisymmetric wave function by asingle determinant of the orbitals Sections 6 and 7 introducethe KohnndashSham formulation whose primary objective is toreplace the interacting electrons by a fictitious but equiva-lent system of noninteracting particles Electrons are reintro-duced as dynamical quantities in Section 8 through the CarndashParrinello formulation Isothermal and isobaric processes areaddressed in this section In Section 9 a path integral for-mulation of molecular dynamics is presented which makesallowance for a quantum treatment of the nuclear degreesof freedom Finally in Section 10 some important compu-tational issues are addressed such as a simplification of the

HindawiAdvances in ChemistryVolume 2018 Article ID 9839641 14 pageshttpsdoiorg10115520189839641

2 Advances in Chemistry

Figure 1 Monoclonal antibody 1F1 isolated from a 1918 influenzapandemic (Spanish Flu) survivor

electronic interaction with the pseudopotential the repre-sentation of orbitals in terms of a functional basis the useof the Fourier and wavelet transform in order to reduce thecomputational complexity and the simulation of larger sys-tems with hybrid molecular dynamics This is followed by aconclusion in Section 11

2 Quantum Molecular Dynamicsand the Schroumldinger EquationA Molecular Perspective

We review some important notions of quantum mechanicsfrom a molecular perspective In quantum mechanics [5ndash7] the Hamiltonian [5ndash7] is the operator corresponding tothe total energy of the molecular system associated with theelectrons and the nuclei The total Hamiltonian is the sumof the kinetic energies of all the particles plus the potentialenergy of the constituent particles in occurrence the elec-trons and the nuclei

H (rR) = minussum119868

ℏ22119872119868

Δ 119868 +H119890 (rR) (1)

where ℏ is the Planck constant and119872119868 is the mass of a givennucleusThe electronic and nuclear Cartesian coordinates fora given particle are defined respectively as

r119894 Š [119903119894119909 119903119894119910 119903119894119911]119879 R119868 Š [119877119868119909 119877119868119910 119877119868119911]119879

(2)

The ensemble of all electronic coordinates is represented byr while the ensemble of all nuclear coordinates is denoted byR In addition the gradient and the Laplacian operators for agiven electron 119894 and nucleus 119868 are defined respectively as

nabla119894 Š120597120597r119894

Δ 119894 Š120597120597r119894 sdot

120597120597r119894

nabla119868 Š120597120597R119868

Δ 119868 Š

120597120597R119868

sdot 120597120597R119868

(3)

The gradient operator a vector is related to the momentumwhile the Laplacian operator a scalar quantity is associatedwith the kinetic energy [5ndash7] The electronic Hamiltonian isdefined as the sum of the kinetic energy of the electrons thepotential energy associated with each pair of electrons thepotential energy associated with each pair of electron-nu-cleus and the potential energy concomitant to each pair ofnuclei

H119890 (rR) Š minussum119894

ℏ22119898119890

Δ 119894 + 141205871205760sum119894lt119895119890210038171003817100381710038171003817r119894 minus r119895

10038171003817100381710038171003817minus 141205871205760sum119868119894

119890211988511986810038171003817100381710038171003817R119868 minus r11989510038171003817100381710038171003817 + 141205871205760sum119868lt119869

11989021198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817

(4)

where 119898119890 is the electronic mass 119890 is the electronic charge119885 is the atomic number (the number of protons in a givennucleus) and 1205760 is the vacuum permittivity The quantummolecular system is characterised by a wave function Φ(rR 119905) whose evolution is determined by the time-dependentSchrodinger equation [5ndash7]

119894ℏ 120597120597119905Φ (rR 119905) = H (rR) Φ (rR 119905) (5)

where 119894 = radicminus1The Schrodinger equation admits a stationaryor time-independent solution

H119890 (rR) Ψ119896 (rR) = 119864119896 (R) Ψ119896 (rR) (6)

where 119864119896(R) is the energy associated with the electronicwave functionΨ119896(rR) and 119896 is a set of quantum numbersthat labelled the Eigenstates or wave functions as well asthe Eigenvalues or energies associated with the stationarySchrodinger equationThe total wave functionΦ(rR 119905)maybe expended in terms of time-dependent nuclear wavefunctions 120594119896(R 119905) and stationary electronic wave functionsΨ119896(rR)

Φ (rR 119905) = infinsum119896=0

120594119896 (R 119905) Ψ119896 (rR) (7)

It should be noticed that this expansion is exact and doesnot involve any approximationThe electronicwave functionsare solution of the stationary Schrodinger equation an Eigenequation which involves the electronic Hamiltonian If onesubstitutes this expansion in the time-dependent Schrodingerequation one obtains a system of equations for the evolutionof the nuclear wave functions

119894ℏ120597120594119896120597119905 = [minussum119868

ℏ22119872119868

Δ 119868 + 119864119896 (R)] 120594119896 +sum119897

C119896119897120594119897 (8)

Advances in Chemistry 3

As may be seen the nuclear wave functions are coupled tothe electronic wave functions The strength of the coupling isdetermined by the coupling coefficients which form amatrix

C119896119897 = int119889rΨlowast119896 [minussum

119868

ℏ22119872119868

Δ 119868]Ψ119897

+sum119868

1119872119868

[int119889rΨlowast119896 [minus119894ℏnabla119868] Ψ119897] [minus119894ℏnabla119868]

(9)

This system of equations is too complex to be solved directlyConsequently in the next subsection we consider two impor-tant approximations of the Schrodinger equation namelythe adiabatic and the BornndashOppenheimer approximationswhich aim to reduce such a complexity These approxima-tions as well as those that later follow reduce substantiallythe duration of the calculations allowing for larger molecularsystems to be simulated and longer time-scales to be explored[8 9]

21 Adiabatic and BornndashOppenheimer Approximations Theabove-mentioned system of differential equations is toocomplex to be solved directly Some approximations must beperformed in order to reduce the computational complexitywhilemaintaining the predictive accuracy of the calculationsThe first approximation which is called the adiabatic approx-imation [5ndash7] assumes that the electronic wave functionsadapt quasi-instantaneously to a variation of the nuclearconfiguration This approximation is justified by the factthat the electrons are much lighter than the nucleus Conse-quently such an approximation is valid unless the motionof the electron becomes relativistic which may happen if theelectrons are too close to the nucleus Mathematically thisapproximation implies that

nabla119868Ψ119897 = 0 (10)

As a result the coupling matrix becomes diagonal

C119896119897 = [minussum119868

ℏ22119872119868

int119889rΨlowast119896 Δ 119868Ψ119896] 120575119896119897 (11)

where 120575119896119897 is the well-known Kronecker deltaThe second approximation which is called the Bornndash

Oppenheimer approximation [6 10 11] assumes that theelectronic and the nuclear motions are separable as a result ofthe difference between nuclear and electronic masses As aresult the expansion reduces to a single Eigen state

Φ (rR 119905) asymp Ψ119896 (rR) 120594119896 (R 119905) (12)

In addition as the energy associated with the wave functionis usually much larger than the corresponding coupling con-stant

C119896119896 ≪ 119864119896 (13)

the time-dependent nuclear Schrodinger further reduces to

119894ℏ120597120594119896120597119905 = [minussum119868

ℏ22119872119868

Δ 119868 + 119864119896 (R)] 120594119896 (14)

Furthermore it is assumed that the electronic wave functionis in its ground or nonexcited state

119896 = 0 997904rArrH119890Ψ0 = 1198640Ψ0 (15)

This occurs if the thermal energy is lower than the energydifference between the ground state and the first excited statethat is if the temperature is low

In the next section we introduce another approximationin which the motion of heavy nuclei is described by a semi-classical equation

3 Ehrenfest Molecular Dynamics

Another possible approximation is to assume that the nuclearmotion is semiclassical as it is the case when the nucleiare relatively heavy This implies that instead of beingdetermined by the Schrodinger equation the average nuclearmotion is determined by Newtonrsquos equation The right formfor this equation is given by the Ehrenfest theorem [6 12ndash14] which states that the potential which governs the classicalmotion of the nucleons is equal to the expectation or averagein the quantum sense of the electronic Hamiltonian withrespect to the electronic wave function

119872119868R119868 (119905) = minusnabla119868 ⟨Ψ 1003816100381610038161003816H1198901003816100381610038161003816 Ψ⟩ = minusnabla119868 ⟨H119890⟩Ψ (16)

where the brandashket notation is to be understood as

⟨Ψ |O| Ψ⟩ equiv int119889rΨlowast (r)OΨ (r) (17)

for any operator O and where R119868 equiv 1198892R1198681198891199052 Thisequation may be further simplified if the adiabatic and theBornndashOppenheimer approximations are enforced

nabla119868 ⟨Ψ 1003816100381610038161003816H1198901003816100381610038161003816 Ψ⟩ 997904rArr

nabla119868 ⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ 997904rArr⟨Ψ0

1003816100381610038161003816nabla119868H1198901003816100381610038161003816 Ψ0⟩

(18)

In the specific context of Ehrenfest molecular dynamics theelectrons follow a time-dependent Schrodinger equation

119894ℏ 120597120597119905Ψ = H119890Ψ (19)

The fact that this equation is time-dependent ensures that theorthogonality of the orbitals is maintained at all times Themotivation is to be found in the fact that the electronicHamil-tonian is aHermitian operator [6] As stated earlier this is notthe case in the BornndashOppenheimer approximation in whichthe electronic wave function is governed by the stationarySchrodinger equationwhich does notmaintain the orthonor-mality over time

In the next section we apply the adiabatic and BornndashOppenheimer approximation in the context of ab initio mo-lecular dynamics


4 BornndashOppenheimer Molecular Dynamics

In BornndashOppenheimer molecular dynamics [6 10 11] itis assumed that the adiabatic and the BornndashOppenheimerapproximations are valid and that the nuclei follow a semi-classical Newton equation whose potential is determinedby the Ehrenfest theorem It is further assumed that theelectronic wave function is in its ground state (lowest energy)Since the stationary Schrodinger equation is used for the elec-tronic degrees of freedom the orthogonality condition mustbe enforcedwith a Lagrangemultiplier as this condition is notpreserved by the stationary Schrodinger equation over timeThe orthogonality of the orbitals 120595119894 is a physical requirementwhich must be enforced at all time in order to be able tocompute real observable quantities The BornndashOppenheimermolecular dynamics is characterised by a Lagrangian [6]which is defined as the difference between the kinetic and thepotential energy

LBO = 12sum

119868

119872119868R2119868 minus ⟨Ψ0

1003816100381610038161003816H1198901003816100381610038161003816 Ψ0⟩

+sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (20)

The first term of the Lagrangian corresponds to the kineticenergy of the nuclei the second term corresponds to thenuclear potential as obtained from the Ehrenfest theoremand the last term is a Lagrange function which ensures thatthe orbitals remain orthonormal at all time The Lagrangemultipliers are denoted by Λ 119894119895 Since the electrons followa FermindashDirac statistics [6] they obey the Pauli ExclusionPrinciple (two electrons cannot be in the same quantumstate) which means that the electronic wave function must bean antisymmetric function of its orbitals namely the wavefunctions associated with the individual electrons

The equations of motion associated with this Lagrangianare obtained from the EulerndashLagrange equations [6] Thereis one system of EulerndashLagrange equations for the nucleardegrees of freedom

119889119889119905 120597L120597R119868

minus 120597L120597R119868

= 0 (21)

and one system of EulerndashLagrange equations for their elec-tronic counterpart

119889119889119905 120575L120575 ⟨1198941003816100381610038161003816 minus

120575L120575 ⟨1205951198941003816100381610038161003816 = 0 997904rArr

119889119889119905 120575L120575lowast119894 (r) minus

120575L120575120595lowast119894 (r) = 0

(22)

From the EulerndashLagrange equations one obtains

119872119868R119868 (119905) = minusnabla119868min120595

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ (23)

for the nuclear equations of motion and

H119890120595119894 (r) = sum119895

Λ 119894119895120595119894 (r) (24)

for the electronic equations of motion One should notice thepresence of the Lagrange multiplier in the electronic equa-tions of motion which ensures that the orbital remains ortho-normal at all time

In the next section we introduce another approximationin order to further reduce the complexity of the electronicHamiltonian

5 HartreendashFock Molecular Dynamics

From now on we shall adopt the atomic unit system

119898119890 = 119890 = ℏ = 141205871205760 = 1 (25)

in order to alleviate the notation In HartreendashFock moleculardynamics [15] the antisymmetric electronic wave functionis approximated by a single determinant of the electronicorbitals 120595119894(r119894)

Ψ = 1radic119873 det[[[[[[[

1205951 (r1) 1205951 (r2) sdot sdot sdot 1205951 (r119873)1205952 (r1) 1205952 (r2) sdot sdot sdot 1205952 (r119873) d

120595119873 (r1) 120595119873 (r119873) sdot sdot sdot 120595119873 (r119873)

]]]]]]] (26)

Two operators are associated with the electronic interactionThe first one the Coulomb operator corresponds to theelectrostatic interaction between the orbitals

J119895 (r) 120595119894 (r) equiv [int119889r1015840120595lowast119895 (r1015840) 11003817100381710038171003817r minus r10158401003817100381710038171003817120595119895 (r1015840)]120595119894 (r) (27)

The second operator the exchange operator [6 15] is associ-ated with the exchange energy a quantum mechanical effectthat occurs as a result of the Pauli Exclusion Principle

K119895 (r) 120595119894 (r) equiv [int119889r1015840120595lowast119895 (r1015840) 11003817100381710038171003817r minus r10158401003817100381710038171003817120595119894 (r1015840)]120595119895 (r) (28)

From the Coulomb and the exchange operators an electronicHamiltonian may be inferred

HHF119890 = minus12Δ + 119881ext (r) + 2sum

119895

J119895 (r) minus sum119895

K119895 (r)equiv 12Δ + 119881HF (r)

(29)

where the external potential is defined as

119881ext (r) Š minussum119868

1198851198681003817100381710038171003817R119868 minus r1003817100381710038171003817 + sum119868lt119869

1198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817 (30)

This Hamiltonian determines in turn the electronicLagrangian

LHF119890 = minus⟨Ψ0

10038161003816100381610038161003816HHF119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (31)


A Lagrange multiplier term is added to the Lagrangian inorder to ensure that the orbitals remain orthonormal at alltime a quantummechanical requirement [6]From theEulerndashLagrange equations one obtains the equations of motion forthe electrons

HHF119890 120595119894 = sum

119895

Λ 119894119895120595119894 (32)

which differ from (24) by the HamiltonianIn the next subsection we seek to replace the interacting

electrons by a fictitious but equivalent system of noninter-acting particles in order to further reduce the computationalcomplexity

6 KohnndashSham Molecular Dynamics

The KohnndashSham formulation [16 17] aims to replace theinteracting electrons by a fictitious system of noninteractingparticles that generate the same electronic density as thephysical systemof interacting particlesThe electronic densityis defined as

119899 Š sum119894

119891119894 ⟨120595119894 | 120595119894⟩ (33)

where 119891119894 is the occupation number of a given orbitalIn this formulation of AIMD the Ehrenfest term is

replaced by the KohnndashSham energy

minΨ0

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ = min120595

119864KS [120595] (34)

The latter is define as

119864KS [119899] Š minus 12119873sum119894=1

119891119894 ⟨120595119894 |Δ| 120595119894⟩ + int119889r119881ext (r) 119899 (r)+ 12 int119889r119881119867 (r) 119899 (r) + 119864XC [119899]

(35)

where the first term is the total electronic kinetic energy asso-ciated with the electrons the second term is the electrostaticsinteraction energy between the electronic density and theexternal potential and the third term is the self-electrostaticinteraction energy associated with the electronic densityThelatter involves the interaction of the electronic density with itself-created electrostatic potential This potential called theHartree potential is obtained by solving the Poisson equation[18]

Δ119881119867 = minus4120587119899 997904rArr119881119867 (r) = int119889r1015840 119899 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817

(36)

The last term is the celebrated exchange-correlation energyor density functional [16 19ndash21] which takes into account theresidual electronic interaction that is the self-interaction ofthe electrons Unfortunately the density functional has noclosed-form solution but many approximations are known

Some of these approximations are considered in the next sec-tion

From the exchange-correlation energy one may definethe exchange-correlation potential

119881XC (r) Š120575119864XC [119899]120575119899 (r) (37)

which is the functional derivative [22] of the exchange-corre-lation energy with respect to the electronic density In addi-tion one may define the KohnndashSham potential

119881KS Š 119881119867 + 119881ext + 119881XC (38)

As for the other approaches a Lagrangian is defined in whichthe orthonormality conditions are enforced with Lagrangemultipliers The electronic equations of motion which aredetermined by the EulerndashLagrange equations are

HKS119890 120595119894 = sum

119895

Λ 119894119895120595119895 (39)

where the fictitious one-particle Hamiltonian is given by

HKS119890 equiv minus12Δ + 119881KS (40)

A canonical formmay be obtained if a unitary transformationis applied to the previous equation

HKS119890 120595119894 = 120576119894120595119894 (41)

In the next section we introduce some density functionals inorder to replace the interacting electrons by an equivalent butyet simpler system of noninteracting particles

7 Exchange-Correlation Energy

The detailed analysis of density functionals also known asexchange-correlation energies [23] is beyond the scope ofthis review We refer the interested reader to [21] for anexhaustive analysis In this section we briefly introduce a fewcommondensity functionals Asmentioned earlier the aimofthe density functional is to express the electronic interactionin terms of the sole electronic density

In the simplest case the exchange-correlation energy is afunctional of the electronic density alone for which the mostimportant representative is the local density approximation

119864LDAXC [119899] = minus34 ( 3120587)

13 int119889r 11989943 (42)

Onemay take into consideration in addition to the electronicdensity the gradient of the electronic density in which casethe approach is called the generalised gradient approxima-tion

119864GGAXC [119899] = int119889r 11989943119865XC (120577) (43)

where the dimensionless reduced gradient is defined as

120577 Š nabla11989911989943 equiv 2 (31205872)13 119904 (44)


Among these approximations is the B88 approximation

119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)

where 120590 refers to the spin and the Perdew Burke and Ernzer-hof (PBE) approximation

119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)

The exchange energy which is associated with the PauliExclusion Principle may be characterised by the HartreendashFock energy

119864HFXC = minus12sum

119894119895

int119889r 119889r1015840120595lowast119894 (r) 120595119895 (r) 120595lowast

119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)

which was introduced earlier These density functionals maybe linearly combined in order to increase their precision suchas in the case of the B3 approximation

119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)

where 119886 119887 and 119888 are empirical parameters Obviously theseparameters are application dependent

Recently it has been proposed to evaluate the functionaldensity with machine learning techniques The functionaldensity is learned by examples instead of directly solving theKohnndashSham equations [17 24 25] As a result substantiallyless time is required to complete the calculations allowing forlarger system to be simulated and longer time-scales to beexplored

In the next section we further improve the precisionof the calculations by reintroducing the dynamic electronicdegrees of freedom which until now have been absent fromthe equations of motion

8 CarndashParrinello Molecular Dynamics

81 Equations of Motion The KohnndashSham dynamics as for-mulated in the previous sections does not take the dynamicsof the electrons into account despite the fact that it is presentan unattractive unphysical feature Indeed one obtains fromthe Lagrangian and the EulerndashLagrange equations

119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)

which means the orbitals are not dynamical fields of themolecular system Nevertheless the Lagrangian may bemodified in order to also include the dynamic nature ofelectronic degrees of freedom through the introduction ofa fictitious electronic kinetic energy term The extended

Lagrangian which is called the CarndashParrinello Lagrangian[11 26 27]

LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)

differs from the original KohnndashSham Lagrangian by thesecond termwhich associates a fictitious kinetic energy to theelectronic orbitals The parameter 120583 acts as a fictitious elec-tronicmass or inertia As in the previous sections the nuclearequations of motion

119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)

and the electronic equations of motion

120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)

may be inferred from the EulerndashLagrange equations Theseequations involve a second order time derivative of theorbitalswhichmeans that the latter are nowproper dynamicalquantities

In the next two subsections we consider ab initiomolecular dynamics at constant temperature and at constantpressure

82 Massive Thermostating and Isothermal Processes In theprevious sections we have implicitly assumed that themolec-ular system under consideration was isolated Of course thisis not compatible withmost experimental conditions [27ndash29]in which the system is either kept at constant temperature(isothermal) in a heat bath or at constant pressure (isobaric)as a consequence for instance of the atmospheric pressureIn this subsection and in the next we explain how to addressthese important issues

As we have recently explained in a computational reviewonmolecular dynamics [30] an isothermal process cannot beobtained by adding an extra term to the Lagrangian Ratherthe equations of motionsmust bemodified directly as a resultof the non-Hamiltonian nature of the isothermal process [31]In order to obtain an isothermal molecular system a frictionterm must be recursively added to each degree of freedom[28 29] Therefore the electronic equations of motion mustbe modified as follows

120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)


while the nuclear equations of motion must take a similarform

119872119868R119868 = minusnabla119868119864KS minus1198721198681205851R119868

11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1

1198991205852119896minus1 minus 1120573] minus 119876119896

119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)

where 120573 equiv 1119896119861119879 119879 is the temperature of the heat bath119896119861 is the Boltzmann constant the 120578119897 are frictional electronicdegrees of freedom the 119876119897

119890 are the friction coefficientsassociated with the electronic orbitals the 120585119896 are frictionalnuclear degrees of freedom the119876119896

119899 are the friction coefficientsassociated with the nuclei and 119892 denotes the number ofdynamical degrees of freedom to which the nuclear thermo-stat chain is coupled The friction terms are proportional tothe velocity of the corresponding degrees of freedom as itis customary Not only must friction terms be added to thephysical degrees of freedom but each nonphysical frictionterm in turn must be thermalised by another nonphysicalfriction term until an isothermal state is properly achievedThe terms 1205781 and 1205851 may be considered as dynamical frictioncoefficients for the physical degrees of freedom

In the next section we present an alternative approachbased on the generalised Langevin equation

83 Generalised Langevin Equation and Isothermal ProcessesMassive thermostating is not the sole approach to simulatean isothermal process Indeed the latter may be achieved bymeans of the generalised Langevin equation [32ndash34] In thissubsection we restrict ourselves to one generalised coordi-nate in order not to clutter the notation The generalisationto more than one coordinate is immediate The generalisedLangevin equation which is a differential stochastic equationmay be written as

119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)

where A119901 is the drift matrix

A119901 Š [119886119901119901 a119879119901a119901 A

] (56)

B119901 is the diffusion matrix 119902 is a generalised coordinateassociatedwith an atomor a nucleus (position)119901 is the corre-sponding generalised momentum and p is a set of 119899p hiddennonphysical momenta The structure of the diffusion matrixB119901 is similar to the one of the corresponding driftmatrixTherandom process is characterised by an uncorrelated Gaussiannoise

⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)

The hidden momenta associated with the generalised Lan-gevin equation may be marginalised because the evolutionof the variables [119901p]119879 in the free particle limit is describedby a linear Markovian stochastic differential equation As aresult the generalised Langevin equation becomes equivalentto a Langevin equation with memory kernel and noise corre-lation [32ndash34]

119902 = 119901 = minus1198811015840 (119902) minus int119905

minusinfin119870 (119905 minus 119904) 119901 (119904) 119889119904 + 120577 (58)

The memory kernel which is a dissipative term associatedwith friction is given by the expression

119870 (119905) = 2119886119901119901120575 (119905) minus a119879119901exp [minus |119905|A] a119901 (59)

whereas the noise correlation which characterised the fluc-tuations associated with the random noise is provided by

119867(119905) Š ⟨120577 (119905) 120577 (0)⟩= 119889119901119901120575 (119905) + a119879119901 exp [minus |119905|A] (Za119901 minus d119901) (60)

where the matrixes Z andD119901 are defined as

Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)

respectively The canonical isothermal ensemble is obtainedby applying the fluctuation-dissipation theorem [35] Thefluctuation-dissipation theorem states that the Fourier trans-formsF of the noise correlation and thememory kernelmustbe related by

(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)

The fluctuation-dissipation theorem implies in turn that

D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)

Therefore the fluctuation-dissipation theorem fixes the dif-fusionmatrix once the driftmatrix is determined As a resultan isothermal process follows immediately

In the next subsection we address isobaric molecularprocesses which are very common as most experiments areperformed at atmospheric pressure

84 Isobaric Processes As opposed to the isothermal processthe isobaric process is a Hamiltonian process which meansthat it may be obtained by adding extra terms to the Lagran-gian [28 36]

In order to model an isobaric molecular process thevolume occupied by the molecular system is divided intocongruent parallelepipeds called Bravais cells These cells arecharacterised by three oriented vectors which correspond to


the three primitive edges spanning their volume These vec-tors are concatenated in order to form a Bravais matrix

h = [a1 | a2 | a3] (65)

The Bravais cell is characterised by its volume and by a localmetric

Ω = det hG = hTh (66)

In order to introduce the volume as a dynamic quantity intothe Lagrangian the nuclear and the electronic coordinates arereformulated in terms of the Bravais matrix and of the scaledcoordinates

R119868 = hS119868r119894 = hs119894 (67)

The Bravais matrix and the scale coordinates constitutedistinct degrees of freedom In order to obtain an isobaric sys-tem the CarndashParrinello Lagrangian must be supplementedwith three additional terms which appear on the third line ofthe Lagrangian

L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)

The first term represents the kinetic energy associated withthe scale coordinates One should notice the presence of themetric or quadratic form in the inner product The secondterm represents the kinetic energy associated with the BravaismatrixThematrix norm is the square of the Frobenius normwhile119882 is a fictitiousmass or inertia attributed to the Bravaiscells The last term is associated with the pressure pof thebarostat (ambient pressure) The equations of motion areobtained as usual from the EulerndashLagrange equations Inparticular the EulerndashLagrange equations for the Bravais cellstake the form

119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)

In the next section we present a path integral formulationof ab initio molecular dynamics which allows a quantumformulation of both the electronic and nuclear degrees offreedom

9 Path Integral Molecular DynamicsToward a Quantum Formulation of Nuclei

The ab initio path integral technique is based on the formula-tion of quantum statistical mechanics in terms of Feynman

path integrals Contrarily to the previous approaches thepath integral method allows for a quantum formulationwhich includes in addition to the electronic degrees offreedom their nuclear counterpart Such a formulation isessential for systems containing light nuclei [37 38]

91 Path Integral Formulation The quantum path integralformulation [36 39ndash42] is based on the partition function ofthe quantum statistical canonical ensemble which is definedas the trace of the exponential of the Hamiltonian operator

Z = tr exp [minus120573H] (70)

The partition function describes the statistical properties ofthe molecular system Since the operators associated with theelectrons and the nuclei do not commute the exponential ofthe Hamiltonian operator must be evaluated with the Trotterfactorization

exp[minus120573(minussum119868

ℏ22119872119868

Δ 119868 +H119890)]

= lim119875rarrinfin

(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)

If the completeness relation

int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)

which is an identity operator is recursively inserted into theTrotter formula and if the adiabatic approximation is per-formed

nabla119868Ψ119896 = 0 997904rArr⟨Ψ119896(119904+1) (R(119904+1)) | Ψ119896(119904) (R(119904))⟩ asymp 120575119896(119904+1)119896(119904) (73)

the partition function becomes

Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]

times exp[minus120573 119875sum119904=1

119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)

are the amplitude and the angular frequency associated withthe quantum harmonic oscillatorsThe latter appear as a con-sequence of the path integral formulationThe computational


time 119904 is a discrete evolution parameter associated with theevolution of the molecular system As such it represents aparticular time-slice The path integral associated with thepartition function is a weighted sum over all possible nucleartrajectories The nuclear configuration at a particular time-slice 119904 is provided by the ensemble of all the individual nuclearconfigurations R(119904)

119868 at this specific instant The weightingfunction corresponds to the exponential factorwhich consistsof two terms the first one is related to the harmonic potentialenergy associated with the nuclei while the second is theenergy associated with the electrons

As in the previous sections the BornndashOppenheimerapproximation is legitimate if the thermal energy is muchsmaller than the difference between the electronic groundstate and the excited states

119864119896 minus 1198640 ≫ 119896119861119879 (76)

It follows that the partition function becomes

Z = lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)

From this partition function an extended Lagrangianmay bedefined

LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)

where1198721015840119868 are fictitiousmasses or unphysical auxiliary param-

eters associated with the nuclei whereas their physical massesare identified by 119872119868 One should notice that 119873 times 119875 ficti-tious momenta P(119904)

119868 = 1198721015840119868R

(119904)119868 have been formally introduced

These momenta are required in order to evaluate numericallythe path integral with Monte Carlo techniques [30 43] Thereader should notice that the momenta affect neither thepartition function (up to an irrelevant normalisation factor)nor the physical observables The ground state energy 1198640(R119904)must be evaluated concurrently with the nuclear energy foreach time-slice

The nuclear coordinates are not linearly independentNevertheless they may become linearly independent if theyare expressed in terms of their normal modes The normaldecomposition [39 41 44] is obtained by representing eachcoordinate in terms of complex Fourier series

R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)

The complex Fourier coefficients are further expanded interms of their real and imaginary parts

u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)

As for the electronic structure it may be obtained fromone of the previously introduced approaches For instancefor a CarndashParrinello technique formulated in terms of aKohnndashSham Hamiltonian the path integral Lagrangian innormal coordinates becomes

LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)

where the normal mode masses are defined as

119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)

to which the fictitious normal mode masses are closely re-lated

1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)

where the centroid adiabaticity parameter 120574 belongs to theinterval 0 lt 120574 le 1These masses are functions of the compu-tational time All the other quantities were defined earlier

In the next subsection we address the calculation ofphysical observables in the path integral framework

92 Physical Observables An observable [6 7 45] is a dy-namic quantity that may be measured experimentally such asthe average energy or the heat capacity Numerous observ-ables may be obtained directly from the partition func-tion for instance

(i) the expectation of the energy (average energy)

⟨119864⟩ = minus120597 lnZ120597120573 (84)

(ii) the variance of the energy or energy fluctuation

⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)


(iii) the heat or thermal capacity

119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)

(v) the Helmholtz free energy

119860 = minus119896119861119879 lnZ (88)

among others In addition if a small perturbation is applied toa molecular system the expectation of the energy associatedwith this perturbation is

119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)

where 120582 is a small running parameterIn the next subsection we present an isothermal formu-

lation of path integrals

93 CarndashParrinello Path Integral Molecular Dynamics andMassiveThermostating Isothermal processes are essential fortwo reasons The first one which is rather evident is that thesimulation of realistic physical conditions often involves theirsimulation [34]The second reason must be found in a rathersubtle shortcoming of the formalism

The massive thermostating of the path integral [34] isa ldquosine qua nonrdquo condition in order to obtain physicallyrealistic results Indeed the harmonic potential leads toinefficient and nonergodic dynamics when microcanonicaltrajectories are used to generate ensemble averages

In contrast the thermostats generate ergodic canonicalaverages at the expense of introducing sets of auxiliary chainvariables which add to the complexity of the calculation butwhich are nevertheless required for its precision and physicaltrustworthiness

As we saw in Section 82 it is not possible to generate anisothermal process simply by adding extra terms to the La-grangian Rather the equations ofmotionwhich are obtainedfrom the EulerndashLagrange equations must be modifiedaccordingly Therefore friction terms must be added recur-sively to the various degrees of freedom As a result theelectronic equations of motion become

120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)

whereas the nuclear equations of motion transform into

119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)

119868 minus R(119904+1)119868 minus R(119904minus1)

119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1

119899 ( 120585(119904)119896minus1)2 minus 1120573] minus 119876119896119899120585(119904)119896

120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)

The quantities appearing in these equations are all defined inSection 82 One should notice that number of equations isquite large due to the evolution parameter swhich is absentfrom the standard CarndashParrinello formulation as reported inSection 93

In the next section we address some important imple-mentational issues

10 Computational Implementation

In this section we present some important implementationalconsiderations In particular we approximate the electronicinteraction with an effective pseudopotential In additionwe demonstrate that the computational complexity may bereduced if the orbitals are expressed in terms of a suitablefunctional basis While ab initio molecular dynamics isrestricted to small molecules because of its computationalcomplexity the method may be extended to larger systems ifa hybrid approach is followed The latter is introduced in thelast subsection

101 Pseudopotential For many calculations the completeknowledge of the electronic interaction is not essential forthe required precision Consequently for the sake of com-putational efficiency the exact electronic potential may beapproximated by means of an effective potential called thepseudopotential [46ndash48] Moreover the relativistic effectsassociated with the core electrons may be implicitly incorpo-rated into the pseudopotential without recourse to explicitand intricate approaches


A commonly employed pseudopotential the dual-spaceGaussian pseudopotential [46 47] is composed of two partsa local potential and a nonlocal potential

119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)

The local potential which described the local interaction isdefined as

119881119871 (119903) Š minus 119885ion119903 erf ( 119903radic2119903119871) + exp(minus12 ( 119903119903119871)2)

times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)

where erf is the error function while 119903119871 1198621 1198622 1198623 and 1198624

are adjustable empirical parameters The nonlocal potential

VNL (r r1015840)= sum

119894

sumℓ

ℓsum119898=minusℓ

119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)

that describes the nonlocal interaction is defined in terms ofthe complex spherical harmonics functions 119884119898

ℓ (120579 120593) and ofthe Gaussian projectors

119901ℓ119894 (119903) = radic2119903ℓ+2(119894minus1) times 119890minus(12)(119903119903ℓ)2

119903ℓ+4119894minus12ℓradicΓ (ℓ + (4119894 minus 1) 2) (95)

In these equations ℓ is the azimuthal quantumnumber and119898is the magnetic quantum number while Γ is the well-knowngamma function

In the next subsection we explore the advantages ofprojecting the orbitals on a functional basis

102 Orbitals and Basis Functions Any continuous functionsuch as an orbital may be represented as a linear combinationof basis functions

1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)

where 119888119894119895 are the projection coefficients and |119891119895⟩ are thebasis functions Such decompositionmay be either physicallymotivated computationally motivated or both For instanceif physical solutions of the Schrodinger equation are knownit is possible to project the orbitals on these solutions [49ndash51] The most common Schrodinger basis functions are theSlater-type basis functions [49]

119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)

and the Gaussian-type basis functions [10 50]

119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866

m are normalisation constants while mrepresents themagnetic quantumnumbers Once an orbital is

projected on a basis it is entirely determined by its projectioncoefficients Thus the resulting representation is parametricAs a result the determination of the projection coefficients isequivalent to the determination of the orbitals the closer thebasis functions are to the real solution the more efficient thecalculation is This is due to the fact that fewer coefficientsare required to adequately represent the underlying orbitalAn even more realistic representation may be obtained if thebasis functions are centred upon their respective nuclei [13]

119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)

where 120601120583 is an atomic basis function and R119868 is the location ofa given nucleus

The orbitals may also be represented by plane waves [14]

119891PWk (r) = 1radicΩ exp [119894k sdot r] (100)

whereΩ is the volume of the cell associated with the underly-ing discrete grid and k is the wave vector associated with theplane wave From a physical point of view plane waves forman appropriate basis when the orbitals are smooth functionsOtherwise a large number of plane waves are required foran accurate reconstruction of the orbitals Consequently thecomputational burden associated with the parametric rep-resentation becomes rapidly prohibitive Nevertheless if theorbitals are smooth functions one may take advantage of theFourier transform in order to reduce the complexity involvedin the calculations of the derivatives For instance the Lapla-cian of a Fourier transformed function is obtained by multi-plying this function by the square of the module of the wavevector

Δ F997904997904rArr 11989621198962 Fminus1997904997904997904rArr Δ

(101)

whereF is the Fourier transform If the Fourier transform iscalculated with the Fast Fourier Transform algorithm thecomplexity of such a calculation for a 119873 times 119873 times 119873 discretegrid reduces to

119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)

which explains why plane wave basis and the Fouriertransforms are prevalent in ab initio molecular dynamicssimulations The computational efficiency may be furtherimproved if the Fourier transformations are evaluated withgraphics processing units (GPU) as their architecture makesthem particularly suited for these calculations especiallyregarding speed [12]

The orbital functions are not always smooth as a result ofthe strength of the electronic interaction In this particularcase the plane wave basis constitutes an inappropriate choiceas a very large number of basis elements are required inorder to describe the quasi-discontinuities Nevertheless onewould be inclined to retain the low computational complexity


associated with plane waves and the Fourier transform whilebeing able to efficiently describe the rougher parts of theorbitals This may be achieved with a wavelet basis and thewavelet transforms [20 48 52ndash54] As opposed to the planewave functions which span the whole space the wavelets arespatially localised Therefore they are particularly suited forthe description of discontinuities and fast-varying functionsIn addition the wavelet basis provides a multiresolution rep-resentation of the orbitals which means that the calculationsmay be performed efficiently at the required resolution levelThe wavelet functions are filter banks [55] In one dimensionthey involve two functions

(i) the scaling function

120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)

which is a high-pass filter responsible for the multiresolutionaspect of the transform

(ii) the mother wavelet

120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)

which is a band-pass filter responsible for the basis localisa-tion The coefficients of the scaling and the mother waveletfunction are related by

119892119894 = minus1119894ℎminus119894+1 (105)

In three dimensions at the lowest resolution the scalingfunction is given by

120601119894119895119896 (r) = 120601 (119909119886 minus 119894) 120601 (119910119886 minus 119895) 120601 (119911119886 minus 119896) (106)

while the mother wavelets become

1205811119894119895119896 (r) = 120581 (119909119886 minus 119894) 120601 (119910119886 minus 119895) 120601 (119911119886 minus 119896) 1205812119894119895119896 (r) = 120601 (119909119886 minus 119894) 120581 (119910119886 minus 119895) 120601 (119911119886 minus 119896) 1205813119894119895119896 (r) = 120581 (119909119886 minus 119894) 120581 (119910119886 minus 119895) 120601 (119911119886 minus 119896) 1205814119894119895119896 (r) = 120601 (119909119886 minus 119894) 120601 (119910119886 minus 119895) 120581 (119911119886 minus 119896) 1205815119894119895119896 (r) = 120581 (119909119886 minus 119894) 120601 (119910119886 minus 119895) 120581 (119911119886 minus 119896) 1205816119894119895119896 (r) = 120601 (119909119886 minus 119894) 120581 (119910119886 minus 119895) 120581 (119911119886 minus 119896) 1205817119894119895119896 (r) = 120581 (119909119886 minus 119894) 120581 (119910119886 minus 119895) 120581 (119911119886 minus 119896)

(107)

where 119886 is the resolution of the underlying computationalgrid As a result any orbital function may be approximatedby a truncated expansion

120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)

where 119904119894119895119896 and 119889]119894119895119896 are the wavelet coefficients Naturallyhigher resolutionsmay be achieved themaximum resolutionis determined by the resolution of the computational gridAs for the Fourier transform the wavelet transform admitsFast Wavelet Transform implementation which has the samecomplexity as its Fourier counterpart

In the next subsection we briefly present a hybrid ap-proachwhich involves both ab initiomolecular dynamics andmolecular dynamics

103 Hybrid QuantumMechanicsndashMolecular DynamicsMeth-ods Because of its inherent complexity the applicability ofab initio molecular dynamics is essentially limited to smallmolecular domains Molecular dynamics which we recentlyreviewed in [30] is suitable for larger molecular systemA drawback is that it is not adapted for the simulation ofchemical complexes since it is not based on first principlesas its quantummechanics counterpart Rather molecular dy-namics relies on empirical potentials [56] and classicalmechanics which allow for the simulation ofmuch largermo-lecular complexes

Therefore in order to simulate larger systems hybridapproaches [12 57ndash59] must be followed In such a method asmall region of interest is simulated with ab initio moleculardynamics while the rest of the molecular system is approxi-mated with molecular mechanics

LQMMM = L

QMCP +L

MM +LQM-MM (109)

An extra Lagrangian term is required in order to ensure aproper coupling between the quantum degrees of freedomand the classical degrees of freedomThis Lagrangian consistsof a bounded and an unbounded partThe bounded part con-sists of the stretching bending and torsional terms which arecharacterised by their distances angles and dihedral anglesrespectively The unbounded part contains the electrostaticinteraction between the molecular mechanics atoms and thequantum density as well as the steric interaction which maybe approximated for instance by a van der Waals potential

11 Conclusions

In this paper we have presented a comprehensive but yetconcise review of ab initio molecular dynamics from a com-putational perspective and from first principles Althoughit is always hazardous to speculate about the future weforesee two important breakthroughs Fourier transformswhich constitute an essential part of many molecular simula-tions may be evaluated with high performance on graphicalprocessing units or GPU [54] The same remark applies tothe wavelet transform whose role is essential in describingdiscontinuous orbitals [52] This opens the door to high per-formance simulations while tempering the limitations asso-ciated with computational complexity

For decades one of the main bottlenecks of molecularsimulations has been the calculation of the density func-tionals As in numerous fields machine learning constitutesa promising avenue for the fast and efficient evaluation ofthese functionals without recourse to explicit calculations


[17 24 25 60] Here the density functional is simply learnedfrom existing examples with machine learning techniquesThis paves the way for the simulation for larger and morecomplex molecular systems We plan to review machinelearning-based approaches in a future publication

Conflicts of Interest

The authors declare that there are no conflicts of interest re-garding the publication of this paper

References

[1] P Carloni U Rothlisberger and M Parrinello ldquoThe role andperspective of ab initio molecular dynamics in the study ofbiological systemsrdquo Accounts of Chemical Research vol 35 no6 pp 455ndash464 2002

[2] L Monticelli and E Salonen Eds Biomolecular SimulationsndashMethods and Protocols Humana Press Springer London UK2013

[3] T D Kuhne Ab-Initio Molecular Dynamics 2013 httpsarxivorgabs12015945

[4] M E Tuckerman ldquoAb initio molecular dynamics Basic con-cepts current trends and novel applicationsrdquo Journal of PhysicsCondensed Matter vol 14 no 50 pp R1297ndashR1355 2002

[5] J Broeckhove and L Lathouwers Eds Time-Dependent Quan-tum Molecular Dynamics NATO Advanced Research Workshopon Time Dependent Quantum Molecular Dynamics Theory andExperiment Springer Snowbird USA 1992

[6] F Jensen Introduction to Computational Chemistry WileyChichester Uk 2017

[7] M E Tuckerman Tuckerman Statistical MechanicsTheory andMolecular Simulation Oxford University Press Oxford UK2013

[8] R P Steele ldquoMultiple-Timestep ab Initio Molecular DynamicsUsing an Atomic Basis Set PartitioningrdquoThe Journal of PhysicalChemistry A vol 119 no 50 pp 12119ndash12130 2015

[9] N Luehr T E Markland and T J Martınez ldquoMultiple timestep integrators in ab initio molecular dynamicsrdquo The Journalof Chemical Physics vol 140 no 8 Article ID 084116 2014

[10] H B Schlegel Li X J M Millam G A Voth G E Scuseriaand M J Frisch ldquoAb initio molecular dynamics Propagatingthe density matrix with Gaussian orbitals III Comparison withBornOppenheimer dynamicsrdquo Journal of Chemical Physics vol117 no 19 pp 9758ndash9763 2001

[11] T D KuhneM Krack F RMohamed andM Parrinello ldquoEffi-cient and accurate car-parrinello-like approach to born-oppen-heimer molecular dynamicsrdquo Physical Review Letters vol 98no 6 Article ID 066401 2007

[12] X Andrade A Castro D Zueco et al ldquoModified ehrenfest for-malism for efficient large-scale ab initio molecular dynamicsrdquoJournal of Chemical Theory and Computation vol 5 no 4 pp728ndash742 2009

[13] M Vacher D Mendive-Tapia M J Bearpark and M A RobbldquoThe second-order Ehrenfest method A practical CASSCFapproach to coupled electron-nuclear dynamicsrdquo TheoreticalChemistry Accounts vol 133 no 7 pp 1ndash12 2014

[14] F Ding J J Goings H Liu D B Lingerfelt and X Li ldquoAb initiotwo-component Ehrenfest dynamicsrdquo The Journal of ChemicalPhysics vol 143 no 11 Article ID 114105 2015

[15] K Kitaura and K Morokuma ldquoA new energy decompositionscheme for molecular interactions within the Hartree-Fockapproximationrdquo International Journal of Quantum Chemistryvol 10 no 2 pp 325ndash340 1976

[16] H S Yu X He and D G Truhlar ldquoMN15-L A New Local Ex-change-Correlation Functional for Kohn-Sham Density Func-tional Theory with Broad Accuracy for Atoms Molecules andSolidsrdquo Journal of ChemicalTheory andComputation vol 12 no3 pp 1280ndash1293 2016

[17] F Brockherde LVogt L LiM E TuckermanK Burke andK-RMuller ldquoBy-passing the Kohn-Sham equations withmachinelearningrdquo Nature Communications vol 8 no 1 article no 8722017

[18] A P Selvadurai Partial Differential Equations in Mechanics 2Springer Berlin Heidelberg Berlin Heidelberg 2000

[19] E Engel and R M Dreizle Density Functional Theory AnAdvanced Course Springer London UK 2011

[20] T D Engeness and T A Arias ldquoMultiresolution analysis forefficient high precision all-electron density-functional calcu-lationsrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 65 no 16 pp 1ndash10 2002

[21] A J Cohen PMori-Sanchez andWYang ldquoChallenges for den-sity functional theoryrdquoChemical Reviews vol 112 no 1 pp 289ndash320 2012

[22] H Ou-Yang and M Levy ldquoTheorem for functional derivativesin density-functional theoryrdquo Physical ReviewA AtomicMolec-ular and Optical Physics vol 44 no 1 pp 54ndash58 1991

[23] P Mori-Sanchez A J Cohen and W Yang ldquoSelf-interaction-free exchange-correlation functional for thermochemistry andkineticsrdquoThe Journal of Chemical Physics vol 124 no 9 ArticleID 091102 2006

[24] L Li T E Baker S R White and K Burke ldquoPure density func-tional for strong correlation and the thermodynamic limit frommachine learningrdquo Physical Review B Condensed Matter andMaterials Physics vol 94 no 24 Article ID 245129 2016

[25] F Pereira K Xiao D A R S Latino C Wu Q Zhang and JAires-De-Sousa ldquoMachine Learning Methods to Predict Den-sity Functional Theory B3LYP Energies of HOMO and LUMOOrbitalsrdquo Journal of Chemical Information andModeling vol 57no 1 pp 11ndash21 2017

[26] M E Tuckerman and M Parrinello ldquoIntegrating the Car-Par-rinello equations I Basic integration techniquesrdquo The Journalof Chemical Physics vol 101 no 2 pp 1302ndash1315 1994

[27] J Lach J Goclon and P Rodziewicz ldquoStructural flexibility ofthe sulfur mustard molecule at finite temperature from Car-Parrinello molecular dynamics simulationsrdquo Journal of Haz-ardous Materials vol 306 pp 269ndash277 2016

[28] G Bussi T Zykova-Timan and M Parrinello ldquoIsothermal-isobaric molecular dynamics using stochastic velocity rescal-ingrdquo The Journal of Chemical Physics vol 130 no 7 Article ID074101 2009

[29] J Lahnsteiner G Kresse A Kumar D D Sarma C Franchiniand M Bokdam ldquoRoom-temperature dynamic correlation be-tween methylammonium molecules in lead-iodine based per-ovskites An ab initiomolecular dynamics perspectiverdquo PhysicalReview B Condensed Matter and Materials Physics vol 94 no21 Article ID 214114 2016

[30] E Paquet and H L Viktor ldquoMolecular dynamics monte carlosimulations and langevin dynamics A computational reviewrdquoBioMed Research International vol 2015 Article ID 1839182015


[31] M E Tuckerman Y Liu G Ciccotti and G J Martyna ldquoNon-Hamiltonian molecular dynamics Generalizing Hamiltonianphase space principles to non-Hamiltonian systemsrdquo The Jour-nal of Chemical Physics vol 115 no 4 pp 1678ndash1702 2001

[32] M Ceriotti G Bussi and M Parrinello ldquoColored-noise ther-mostats ala Carterdquo Journal of Chemical Theory and Computa-tion vol 6 no 4 pp 1170ndash1180 2010

[33] M Ceriotti G Bussi and M Parrinello ldquoLangevin equationwith colored noise for constant-temperature molecular dynam-ics simulationsrdquo Physical Review Letters vol 102 no 2 ArticleID 020601 2009

[34] M Ceriotti M Parrinello T E Markland and D E Mano-lopoulos ldquoEfficient stochastic thermostatting of path integralmolecular dynamicsrdquo The Journal of Chemical Physics vol 133no 12 Article ID 124104 2010

[35] R Kubo ldquoThe fluctuation-dissipation theoremrdquo Reports on Pro-gress in Physics vol 29 no 1 article no 306 pp 255ndash284 1966

[36] G JMartyna AHughes andM E Tuckerman ldquoMolecular dy-namics algorithms for path integrals at constant pressurerdquo TheJournal of Chemical Physics vol 110 no 7 pp 3275ndash3290 1999

[37] T Spura C John S Habershon and T D Kuhne ldquoNuclearquantum effects in liquid water from path-integral simulationsusing an ab initio force-matching approachrdquoMolecular Physicsvol 113 no 8 pp 808ndash822 2015

[38] O Marsalek and T E Markland ldquoAb initio molecular dynamicswith nuclear quantum effects at classical cost Ring polymercontraction for density functional theoryrdquoThe Journal of Chem-ical Physics vol 144 no 5 Article ID 054112 2016

[39] M E Tuckerman DMarxM L Klein et al ldquoEfficient and gen-eral algorithms for path integral Car-Parrinello molecular dy-namicsrdquo The Journal of Chemical Physics vol 104 no 14 pp5579ndash5588 1996

[40] P Minary G J Martyna and M E Tuckerman ldquoAlgorithmsand novel applications based on the isokinetic ensemble I Bio-physical and path integral molecular dynamicsrdquoThe Journal ofChemical Physics vol 118 no 6 pp 2510ndash2526 2003

[41] M Ceriotti J More and D E Manolopoulos ldquoI-PI A Pythoninterface for ab initio path integral molecular dynamics simu-lationsrdquo Computer Physics Communications vol 185 no 3 pp1019ndash1026 2014

[42] H Y Geng ldquoAccelerating ab initio path integral molecular dy-namics withmultilevel sampling of potential surfacerdquo Journal ofComputational Physics vol 283 pp 299ndash311 2015

[43] K A Fichthorn and W H Weinberg ldquoTheoretical foundationsof dynamicalMonte Carlo simulationsrdquoThe Journal of ChemicalPhysics vol 95 no 2 pp 1090ndash1096 1991

[44] D Marx and M Parrinello ldquoAb initio path integral moleculardynamics Basic ideasrdquoThe Journal of Chemical Physics vol 104no 11 pp 4077ndash4082 1995

[45] O Marsalek P-Y Chen R Dupuis et al ldquoEfficient calculationof free energy differences associated with isotopic substitutionusing path-integral molecular dynamicsrdquo Journal of ChemicalTheory and Computation vol 10 no 4 pp 1440ndash1453 2014

[46] S Goedecker and M Teter ldquoSeparable dual-space Gaussianpseudopotentialsrdquo Physical Review B Condensed Matter andMaterials Physics vol 54 no 3 pp 1703ndash1710 1996

[47] C Hartwigsen S Goedecker and J Hutter ldquoRelativistic sep-arable dual-space Gaussian pseudopotentials from H to RnrdquoPhysical Review B CondensedMatter andMaterials Physics vol58 no 7 pp 3641ndash3662 1998

[48] L Genovese A Neelov S Goedecker et al ldquoDaubechies wave-lets as a basis set for density functional pseudopotential calcu-lationsrdquo The Journal of Chemical Physics vol 129 no 1 ArticleID 014109 2008

[49] M A Watson N C Handy and A J Cohen ldquoDensity func-tional calculations using Slater basis sets with exact exchangerdquoThe Journal of Chemical Physics vol 119 no 13 pp 6475ndash64812003

[50] X Andrade and A Aspuru-Guzik ldquoReal-space density func-tional theory on graphical processing units Computational ap-proach and comparison to Gaussian basis set methodsrdquo Journalof Chemical Theory and Computation vol 9 no 10 pp 4360ndash4373 2013

[51] S V Levchenko X Ren J Wieferink et al ldquoHybrid functionalsfor large periodic systems in an all-electron numeric atom-centered basis frameworkrdquo Computer Physics Communicationsvol 192 pp 60ndash69 2015

[52] T Deutsch and L Genovese ldquoWavelets for electronic structurecalculationsrdquo Collection SFN vol 12 pp 33ndash76 2011

[53] AH Prociuk and S S Iyengar ldquoAmultiwavelet treatment of thequantum subsystem in quantumwavepacket ab initiomoleculardynamics through an hierarchical partitioning of momentumspacerdquo Journal of ChemicalTheory and Computation vol 10 no8 pp 2950ndash2963 2014

[54] L Genovese B Videau D Caliste J-F Mehaut S Goedeckerand T Deutsch ldquoWavelet-Based Density Functional Theory onMassively Parallel Hybrid Architecturesrdquo Electronic StructureCalculations on Graphics Processing Units From QuantumChemistry to Condensed Matter Physics pp 115ndash134 2016

[55] R C Gonzalez and R EWoodsDigital Image Processing Pear-son Prentice Hall Upper Saddle River NJ USA 2008

[56] K Vanommeslaeghe E Hatcher and C Acharya ldquoCHARMMgeneral force field a force field for drug-likemolecules compati-blewith theCHARMMall-atomadditive biological force fieldsrdquoJournal of Computational Chemistry vol 31 no 4 pp 671ndash6902010

[57] J R Perilla B C Goh C K Cassidy et al ldquoMolecular dy-namics simulations of large macromolecular complexesrdquo Cur-rent Opinion in Structural Biology vol 31 pp 64ndash74 2015

[58] N Sahu and S R Gadre ldquoMolecular tailoring approach A routefor ab initio treatment of large clustersrdquo Chemical Reviews vol47 no 9 pp 2739ndash2747 2014

[59] J Dreyer B Giuseppe E Ippoliti V Genna M De Vivo andP Carloni ldquoFirst principles methods in biology continuummodels to hybrid ab initio quantum mechanicsmolecularmechanicsrdquo in Simulating Enzyme Reactivity ComputationalMethods in Enzyme Catalysis I Tunon and V Moliner Edschapter 9 Royal Society of Chemistry London UK 2016

[60] V Botu and R Ramprasad ldquoAdaptive machine learning frame-work to accelerate ab initio molecular dynamicsrdquo InternationalJournal of Quantum Chemistry vol 115 no 16 pp 1074ndash10832015

TribologyAdvances in

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Submit your manuscripts atwwwhindawicom


Figure 1 Monoclonal antibody 1F1 isolated from a 1918 influenzapandemic (Spanish Flu) survivor

electronic interaction with the pseudopotential the repre-sentation of orbitals in terms of a functional basis the useof the Fourier and wavelet transform in order to reduce thecomputational complexity and the simulation of larger sys-tems with hybrid molecular dynamics This is followed by aconclusion in Section 11

2 Quantum Molecular Dynamicsand the Schroumldinger EquationA Molecular Perspective

We review some important notions of quantum mechanicsfrom a molecular perspective In quantum mechanics [5ndash7] the Hamiltonian [5ndash7] is the operator corresponding tothe total energy of the molecular system associated with theelectrons and the nuclei The total Hamiltonian is the sumof the kinetic energies of all the particles plus the potentialenergy of the constituent particles in occurrence the elec-trons and the nuclei

H (rR) = minussum119868

ℏ22119872119868

Δ 119868 +H119890 (rR) (1)

where ℏ is the Planck constant and119872119868 is the mass of a givennucleusThe electronic and nuclear Cartesian coordinates fora given particle are defined respectively as

r119894 Š [119903119894119909 119903119894119910 119903119894119911]119879 R119868 Š [119877119868119909 119877119868119910 119877119868119911]119879

(2)

The ensemble of all electronic coordinates is represented byr while the ensemble of all nuclear coordinates is denoted byR In addition the gradient and the Laplacian operators for agiven electron 119894 and nucleus 119868 are defined respectively as

nabla119894 Š120597120597r119894

Δ 119894 Š120597120597r119894 sdot

120597120597r119894

nabla119868 Š120597120597R119868

Δ 119868 Š

120597120597R119868

sdot 120597120597R119868

(3)

The gradient operator a vector is related to the momentumwhile the Laplacian operator a scalar quantity is associatedwith the kinetic energy [5ndash7] The electronic Hamiltonian isdefined as the sum of the kinetic energy of the electrons thepotential energy associated with each pair of electrons thepotential energy associated with each pair of electron-nu-cleus and the potential energy concomitant to each pair ofnuclei

H119890 (rR) Š minussum119894

ℏ22119898119890

Δ 119894 + 141205871205760sum119894lt119895119890210038171003817100381710038171003817r119894 minus r119895

10038171003817100381710038171003817minus 141205871205760sum119868119894

119890211988511986810038171003817100381710038171003817R119868 minus r11989510038171003817100381710038171003817 + 141205871205760sum119868lt119869

11989021198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817

(4)

where 119898119890 is the electronic mass 119890 is the electronic charge119885 is the atomic number (the number of protons in a givennucleus) and 1205760 is the vacuum permittivity The quantummolecular system is characterised by a wave function Φ(rR 119905) whose evolution is determined by the time-dependentSchrodinger equation [5ndash7]

119894ℏ 120597120597119905Φ (rR 119905) = H (rR) Φ (rR 119905) (5)

where 119894 = radicminus1The Schrodinger equation admits a stationaryor time-independent solution

H119890 (rR) Ψ119896 (rR) = 119864119896 (R) Ψ119896 (rR) (6)

where 119864119896(R) is the energy associated with the electronicwave functionΨ119896(rR) and 119896 is a set of quantum numbersthat labelled the Eigenstates or wave functions as well asthe Eigenvalues or energies associated with the stationarySchrodinger equationThe total wave functionΦ(rR 119905)maybe expended in terms of time-dependent nuclear wavefunctions 120594119896(R 119905) and stationary electronic wave functionsΨ119896(rR)

Φ (rR 119905) = infinsum119896=0

120594119896 (R 119905) Ψ119896 (rR) (7)

It should be noticed that this expansion is exact and doesnot involve any approximationThe electronicwave functionsare solution of the stationary Schrodinger equation an Eigenequation which involves the electronic Hamiltonian If onesubstitutes this expansion in the time-dependent Schrodingerequation one obtains a system of equations for the evolutionof the nuclear wave functions

119894ℏ120597120594119896120597119905 = [minussum119868

ℏ22119872119868

Δ 119868 + 119864119896 (R)] 120594119896 +sum119897

C119896119897120594119897 (8)




119868

ℏ22119872119868

Δ 119868]Ψ119897

+sum119868

1119872119868


(9)



nabla119868Ψ119897 = 0 (10)


C119896119897 = [minussum119868

ℏ22119872119868

int119889rΨlowast119896 Δ 119868Ψ119896] 120575119896119897 (11)



Φ (rR 119905) asymp Ψ119896 (rR) 120594119896 (R 119905) (12)


C119896119896 ≪ 119864119896 (13)


119894ℏ120597120594119896120597119905 = [minussum119868

ℏ22119872119868

Δ 119868 + 119864119896 (R)] 120594119896 (14)


119896 = 0 997904rArrH119890Ψ0 = 1198640Ψ0 (15)









nabla119868 ⟨Ψ 1003816100381610038161003816H1198901003816100381610038161003816 Ψ⟩ 997904rArr

nabla119868 ⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ 997904rArr⟨Ψ0

1003816100381610038161003816nabla119868H1198901003816100381610038161003816 Ψ0⟩

(18)


119894ℏ 120597120597119905Ψ = H119890Ψ (19)






LBO = 12sum

119868

119872119868R2119868 minus ⟨Ψ0

1003816100381610038161003816H1198901003816100381610038161003816 Ψ0⟩

+sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (20)



119889119889119905 120597L120597R119868

minus 120597L120597R119868

= 0 (21)


119889119889119905 120575L120575 ⟨1198941003816100381610038161003816 minus

120575L120575 ⟨1205951198941003816100381610038161003816 = 0 997904rArr

119889119889119905 120575L120575lowast119894 (r) minus

120575L120575120595lowast119894 (r) = 0

(22)


119872119868R119868 (119905) = minusnabla119868min120595

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ (23)


H119890120595119894 (r) = sum119895

Λ 119894119895120595119894 (r) (24)





119898119890 = 119890 = ℏ = 141205871205760 = 1 (25)


Ψ = 1radic119873 det[[[[[[[



]]]]]]] (26)







119895

J119895 (r) minus sum119895

K119895 (r)equiv 12Δ + 119881HF (r)

(29)



1198851198681003817100381710038171003817R119868 minus r1003817100381710038171003817 + sum119868lt119869

1198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817 (30)



10038161003816100381610038161003816HHF119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (31)



HHF119890 120595119894 = sum

119895

Λ 119894119895120595119894 (32)





119899 Š sum119894

119891119894 ⟨120595119894 | 120595119894⟩ (33)



minΨ0

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ = min120595

119864KS [120595] (34)


119864KS [119899] Š minus 12119873sum119894=1


(35)



(36)




119881XC (r) Š120575119864XC [119899]120575119899 (r) (37)


119881KS Š 119881119867 + 119881ext + 119881XC (38)


HKS119890 120595119894 = sum

119895

Λ 119894119895120595119895 (39)




HKS119890 120595119894 = 120576119894120595119894 (41)





119864LDAXC [119899] = minus34 ( 3120587)

13 int119889r 11989943 (42)


119864GGAXC [119899] = int119889r 11989943119865XC (120577) (43)


120577 Š nabla11989911989943 equiv 2 (31205872)13 119904 (44)



119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)


119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)



119894119895


119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)


119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)






119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)



LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)


119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)


120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)





120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)




11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls







119868

ℏ22119872119868

Δ 119868]Ψ119897

+sum119868

1119872119868


(9)



nabla119868Ψ119897 = 0 (10)


C119896119897 = [minussum119868

ℏ22119872119868

int119889rΨlowast119896 Δ 119868Ψ119896] 120575119896119897 (11)



Φ (rR 119905) asymp Ψ119896 (rR) 120594119896 (R 119905) (12)


C119896119896 ≪ 119864119896 (13)


119894ℏ120597120594119896120597119905 = [minussum119868

ℏ22119872119868

Δ 119868 + 119864119896 (R)] 120594119896 (14)


119896 = 0 997904rArrH119890Ψ0 = 1198640Ψ0 (15)









nabla119868 ⟨Ψ 1003816100381610038161003816H1198901003816100381610038161003816 Ψ⟩ 997904rArr

nabla119868 ⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ 997904rArr⟨Ψ0

1003816100381610038161003816nabla119868H1198901003816100381610038161003816 Ψ0⟩

(18)


119894ℏ 120597120597119905Ψ = H119890Ψ (19)






LBO = 12sum

119868

119872119868R2119868 minus ⟨Ψ0

1003816100381610038161003816H1198901003816100381610038161003816 Ψ0⟩

+sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (20)



119889119889119905 120597L120597R119868

minus 120597L120597R119868

= 0 (21)


119889119889119905 120575L120575 ⟨1198941003816100381610038161003816 minus

120575L120575 ⟨1205951198941003816100381610038161003816 = 0 997904rArr

119889119889119905 120575L120575lowast119894 (r) minus

120575L120575120595lowast119894 (r) = 0

(22)


119872119868R119868 (119905) = minusnabla119868min120595

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ (23)


H119890120595119894 (r) = sum119895

Λ 119894119895120595119894 (r) (24)





119898119890 = 119890 = ℏ = 141205871205760 = 1 (25)


Ψ = 1radic119873 det[[[[[[[



]]]]]]] (26)







119895

J119895 (r) minus sum119895

K119895 (r)equiv 12Δ + 119881HF (r)

(29)



1198851198681003817100381710038171003817R119868 minus r1003817100381710038171003817 + sum119868lt119869

1198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817 (30)



10038161003816100381610038161003816HHF119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (31)



HHF119890 120595119894 = sum

119895

Λ 119894119895120595119894 (32)





119899 Š sum119894

119891119894 ⟨120595119894 | 120595119894⟩ (33)



minΨ0

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ = min120595

119864KS [120595] (34)


119864KS [119899] Š minus 12119873sum119894=1


(35)



(36)




119881XC (r) Š120575119864XC [119899]120575119899 (r) (37)


119881KS Š 119881119867 + 119881ext + 119881XC (38)


HKS119890 120595119894 = sum

119895

Λ 119894119895120595119895 (39)




HKS119890 120595119894 = 120576119894120595119894 (41)





119864LDAXC [119899] = minus34 ( 3120587)

13 int119889r 11989943 (42)


119864GGAXC [119899] = int119889r 11989943119865XC (120577) (43)


120577 Š nabla11989911989943 equiv 2 (31205872)13 119904 (44)



119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)


119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)



119894119895


119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)


119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)






119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)



LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)


119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)


120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)





120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)




11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls







LBO = 12sum

119868

119872119868R2119868 minus ⟨Ψ0

1003816100381610038161003816H1198901003816100381610038161003816 Ψ0⟩

+sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (20)



119889119889119905 120597L120597R119868

minus 120597L120597R119868

= 0 (21)


119889119889119905 120575L120575 ⟨1198941003816100381610038161003816 minus

120575L120575 ⟨1205951198941003816100381610038161003816 = 0 997904rArr

119889119889119905 120575L120575lowast119894 (r) minus

120575L120575120595lowast119894 (r) = 0

(22)


119872119868R119868 (119905) = minusnabla119868min120595

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ (23)


H119890120595119894 (r) = sum119895

Λ 119894119895120595119894 (r) (24)





119898119890 = 119890 = ℏ = 141205871205760 = 1 (25)


Ψ = 1radic119873 det[[[[[[[



]]]]]]] (26)







119895

J119895 (r) minus sum119895

K119895 (r)equiv 12Δ + 119881HF (r)

(29)



1198851198681003817100381710038171003817R119868 minus r1003817100381710038171003817 + sum119868lt119869

1198851198681198851198691003817100381710038171003817R119868 minus R1198691003817100381710038171003817 (30)



10038161003816100381610038161003816HHF119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895) (31)



HHF119890 120595119894 = sum

119895

Λ 119894119895120595119894 (32)





119899 Š sum119894

119891119894 ⟨120595119894 | 120595119894⟩ (33)



minΨ0

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ = min120595

119864KS [120595] (34)


119864KS [119899] Š minus 12119873sum119894=1


(35)



(36)




119881XC (r) Š120575119864XC [119899]120575119899 (r) (37)


119881KS Š 119881119867 + 119881ext + 119881XC (38)


HKS119890 120595119894 = sum

119895

Λ 119894119895120595119895 (39)




HKS119890 120595119894 = 120576119894120595119894 (41)





119864LDAXC [119899] = minus34 ( 3120587)

13 int119889r 11989943 (42)


119864GGAXC [119899] = int119889r 11989943119865XC (120577) (43)


120577 Š nabla11989911989943 equiv 2 (31205872)13 119904 (44)



119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)


119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)



119894119895


119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)


119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)






119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)



LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)


119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)


120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)





120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)




11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls






HHF119890 120595119894 = sum

119895

Λ 119894119895120595119894 (32)





119899 Š sum119894

119891119894 ⟨120595119894 | 120595119894⟩ (33)



minΨ0

⟨Ψ01003816100381610038161003816H119890

1003816100381610038161003816 Ψ0⟩ = min120595

119864KS [120595] (34)


119864KS [119899] Š minus 12119873sum119894=1


(35)



(36)




119881XC (r) Š120575119864XC [119899]120575119899 (r) (37)


119881KS Š 119881119867 + 119881ext + 119881XC (38)


HKS119890 120595119894 = sum

119895

Λ 119894119895120595119895 (39)




HKS119890 120595119894 = 120576119894120595119894 (41)





119864LDAXC [119899] = minus34 ( 3120587)

13 int119889r 11989943 (42)


119864GGAXC [119899] = int119889r 11989943119865XC (120577) (43)


120577 Š nabla11989911989943 equiv 2 (31205872)13 119904 (44)



119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)


119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)



119894119895


119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)


119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)






119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)



LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)


119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)


120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)





120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)




11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls






119865B88XC (120577) = 34 ( 6120587)

13 + 12057312057721205901 + 6120573120577120590sinhminus1120577120590 (45)


119865PBEXC (119904) = 34 ( 3120587)

13 + 12058311990421 + 1205831199042120581minus1 (46)



119894119895


119895 (r1015840) 120595119894 (r1015840)1003817100381710038171003817r minus r10158401003817100381710038171003817 (47)


119864B3XC = 119886119864HF

XC + (1 minus 119886) 119864LDAXC + 119887Δ119864B88

XC + 119888119864GGAXC

+ (1 minus 119888) 119864LDAXC (48)






119889119889119905 120597L120597lowast119894 (r) equiv 0 (49)



LCP = 12sum

119868

119872119868R2119868 +sum

119894

120583 ⟨119894 | 119894⟩ + ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩+sum

119894119895

Λ 119894119895 (⟨120595119894 | 120595119895⟩ minus 120575119894119895)(50)


119872119868R119868 = minusnabla119868 ⟨Ψ0

10038161003816100381610038161003816HKS119890

10038161003816100381610038161003816 Ψ0⟩ +sum119894119895

Λ 119894119895nabla119868 ⟨120595119894 | 120595119895⟩ (51)


120583119894 (119905) = minusHKS119890 120595119894 (119905) + sum

119895

Λ 119894119895120595119895 (119905) (52)





120583 1003816100381610038161003816 119894⟩ = minusHKS119890

1003816100381610038161003816120595119894⟩ +sum119895

Λ 119894119895

10038161003816100381610038161003816120595119895⟩ minus 120583 1205781 1003816100381610038161003816 119894⟩

1198761119890 1205781 = 2[120583sum

119894

⟨119894 | 119894⟩ minus 1198731198902120573119890

] minus 1198761119890 1205781 1205782

119876119897119890 120578119897 = [119876119897minus1

119890 1205782119897minus1 minus 1120573119890

] minus 119876119897119890 120578119897 120578119897+1 (1 minus 120575119897119871)

119897 = 2 119871

(53)




11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls







11987611198991205851 = [sum

119868

119872119868R2119868 minus 119892120573] minus 1198761

1198991205851 1205852

119876119896119899120585119896 = [119876119896minus1


119899120585119896 120585119896+1 (1 minus 120575119896119870)

119896 = 2 119870

(54)






119902 = 119901 [p] = [minus1198811015840 (119902)

0] minus A119901 [119901

p] + B119901 [120585] (55)


A119901 Š [119886119901119901 a119879119901a119901 A

] (56)


⟨120585119894 (119905) 120585119894 (0)⟩ = 120575119894119895120575 (119905) (57)


119902 = 119901 = minus1198811015840 (119902) minus int119905







Z Š intinfin

0119890minusA119905D119890minusA119879119905119889119905 (61)

D119901 Š B119901B119879119901 (62)


(F ∘ 119867) (120596) = 119896119861119879 (F ∘ 119870) (120596) (63)


D119901 = B119901B119879119901 = 119896119861119879 (A119901 + A119879

119901) (64)







h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls






h = [a1 | a2 | a3] (65)




R119868 = hS119868r119894 = hs119894 (67)


L = sum119894

120583 ⟨119894 (hs) | 119894 (hs)⟩ + 119864KS [120595 hS]+sum

119894119895

Λ 119894119895 (⟨120595119894 (hs) | 120595119895 (hs)⟩ minus 120575119894119895)+ 12sum

119868

119872119868 (S119879119868GS119868) + 12119882 tr (h119879h) minus 119901Ω(68)


119889119889119905 120597L120597 (h)

119906V

minus 120597L120597 (h)119906V = 0 (69)









ℏ22119872119868

Δ 119868 +H119890)]


(exp[120573119875sum119868

ℏ22119872119868

Δ 119868] exp [minus120573119875H119890])119875

(71)


int119889Rsum119896

1003816100381610038161003816Ψ119896 (R) R⟩ ⟨R Ψ119896 (R)1003816100381610038161003816 = I (72)




Z = sum119896

lim119875rarrinfin

119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 1119875119864119896 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(74)

The quantities

119860119875 equiv ( 1198721198681198752120587120573ℏ2)32

1205962119875 = 119875ℏ21205732

(75)






119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls








119864119896 minus 1198640 ≫ 119896119861119879 (76)



119875prod119904=1

[ 119873prod119868=1

int]


119873sum119868=1

121198721198681205962119875 (R(119904)

119868 minus R(119904+1)119868 )2 + 11198751198640 (R(119904))]

times 119875prod119904=1

119873prod119868=1

119860119875119889R(119904)119868

(77)


LPI

= 119875sum119904=1

119873sum119868=1

( 121198721015840119868

(P(119904)119868 )2 minus 121198721198681205962

119875 (R(119904)119868 minus R(119904+1)

119868 )2)

minus 11198751198640 (R119904) (78)



119868 = 1198721015840119868R




R(119904)119868 997888rarr 119875sum

119896=1

a(119896)119868 1198902120587119894(119904minus1)(119896minus1)119875 (79)


u(1)119868 = a(1)119868

u(119875)119868 = a((119875+2)2)119868

u(2119904minus2)119868 = Re (a(119904)119868 )

u(2119904minus1)119868 = Im (a(119904)119868 )

(80)


LPICP = 1119875

119875sum119904=1

sum119894

120583 ⟨(119904)119894 | (119904)

119894 ⟩minus 119864KS [120595(119904) (R (u))(119904)]+sum

119894119895

Λ(119904)119894119895 (⟨120595(119904)

119894 | 120595(119904)119895 ⟩ minus 120575119894119895)

+ 119875sum119904=1

12sum119868

1198721015840119868

(119904) (u(119904)119868 )2 minus 12sum

119868

119872(119904)119868 1205962

119875 (u(119904)119868 )2

(81)


119872(1)119868 = 0

119872(119904)119868 = 2119875(1 minus cos 2120587 (119904 minus 1)119875 )119872119868 119904 = 2 119875 (82)


1198721015840119868

(1) = 1198721198681198721015840

119868

(119904) = 120574119872(119904)119868 119904 = 2 119875 (83)





⟨119864⟩ = minus120597 lnZ120597120573 (84)


⟨(Δ119864)2⟩ = 1205972Z1205971205732(85)



119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls






119862V = 1119896119861119879 ⟨(Δ119864)2⟩ (86)

(iv) the entropy

119878 = 120597120597119879 (119896119861119879 lnZ) (87)


119860 = minus119896119861119879 lnZ (88)


119864 = 1198640 + 120582119882 997904rArr⟨119882⟩ = minus 1120573 120597120597120582 lnZ (89)







120583 10038161003816100381610038161003816 (119904)119894 ⟩ = minus120575119864KS [120595(119904)R(119904)]

120575 (120595(119904)119894 )lowast +sum

119895

Λ(119904)119894119895

10038161003816100381610038161003816120595(119904)119894 ⟩

minus 120583 120578(119904)1

10038161003816100381610038161003816120595(119904)119894 ⟩

1198761119890 120578(119904)1 = 2[120583sum

119894

⟨120595(119904)119894 | 120595(119904)

119894 ⟩ minus 1198731198902120573119890

] minus 1198761119890 120578(119904)1 120578(119904)2

119876119897119890 120578(119904)119897 = [119876119897minus1

119890 ( 120578(119904)119897minus1)2 minus 1120573119890

]minus 119876119897

119890 120578(119904)119897120578(119904)119897+1 (1 minus 1205752119871)

119897 = 2 119871 119904 = 1 119875(90)


119872119868R(119904)119868 = minus 1119875

120597119864KS [120595(119904)R(119904)]120597R(119904)

119868

minus1198721198681205962119875 (2R(119904)


119868 )minus1198721015840

119868120585(119904)1 R(119904)

119868 1198761

119899120585(119904)1 = [sum

119868

1198721015840119868 (R(119904)

119868 )2 minus 119892120573] minus 1198761119899120585(119904)1

120585(119904)2 119876119896

119899120585(119904)119896 = [119876119896minus1


120585(119904)119896+1 (1 minus 120575119896119870) 119896 = 2 119870 119904 = 1 119875

(91)








119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls






119881PP (r r1015840) = 119881119871 (119903) + 119881NL (r r1015840) (92)



times (1198621 + 1198622 ( 119903119903119871)2 + 1198623 ( 119903119903119871)

4 + 1198624 ( 119903119903119871)6)

(93)



VNL (r r1015840)= sum

119894

sumℓ


119884119898ℓ (120579 120593) 119901ℓ

119894 (119903) 119881ℓ119894 119901ℓ

119894 (119903) (119884119898ℓ )lowast (1205791015840 1205931015840) (94)








1003816100381610038161003816120595119894⟩ = sum119895

119888119894119895 10038161003816100381610038161003816119891119895⟩ (96)


119891119878m (r) = 119873119878

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120577m r] (97)


119891119866m (r) = 119873119866

m119903119898119909119909 119903119898119910119910 119903119898119911119911 exp [minus120572m r2] (98)

where 119873119878m and 119873119866



119891119894 (r) = sum120583119868

119888119868119894120583120601120583 (r minus R119868) (99)






(101)


119874(1198736)10038161003816100381610038161003816FT 997904rArr 119874((119873 log119873)3)10038161003816100381610038161003816FFT (102)






120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls







120601 (119909) = radic2 119898sum119894=1minus119898

ℎ119895120601 (2119909 minus 119894) (103)



120581 (119909) = radic2 119898sum119894=1minus119898

119892119895120581 (2119909 minus 119894) (104)


119892119894 = minus1119894ℎminus119894+1 (105)





(107)


120595 (r) = sum119894119895119896

119904119894119895119896120601119894119895119896 (r) + sum119894119895119896

7sum]=1

119889]119894119895119896120581]119894119895119896 (r) (108)





LQMMM = L

QMCP +L

MM +LQM-MM (109)


11 Conclusions







References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls








References



































































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls








































Journal of

Chemistry





Journal of

Volume 2018






Volume 2018




Journal of






Enzyme Research


Journal of



MaterialsJournal of






Na

nom

ate

ria

ls




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