Introduction to Computational Chemistrycct.lsu.edu/~apacheco/tutorials/Intro2CompChem.pdf ·...

Introduction to Computational Chemistry

Alexander B. Pacheco

User Services ConsultantLSU HPC & [email protected]

LONI Worshop SeriesTulane University, New Orleans

March 25, 2011

High Performance Computing @ Louisiana State University - http://www.hpc.lsu.edu March 25, 2011 1/66

Outline

1 Introduction

2 Ab Initio Methods

3 Density Functional Theory

4 Semi-empirical Methods

5 Basis Sets

6 Molecular Mechanics

7 Quantum Mechanics/Molecular Mechanics (QM/MM)

8 Molecular Dynamics

9 Computational Chemistry Programs

10 Example Jobs


What is Computational Chemistry

Computational Chemistry is a branch of chemistry thatuses computer science to assist in solving chemicalproblems.Incorporates the results of theoretical chemistry intoefficient computer programs.Application to single molecule, groups of molecules, liquidsor solids.Calculates the structure and properties of interest.Computational Chemistry Methods range from

1 Highly accurate (Ab-initio,DFT) feasible for small systems2 Less accurate (semi-empirical)3 Very Approximate (Molecular Mechanics) large systems


What is Computational Chemistry

Theoretical Chemistry: broadly can be divided into twomain categories

1 Static Methods⇒ Time-Independent Schrödinger Equation

♦ Quantum Chemical/Ab Initio /Electronic Structure Methods♦ Molecular Mechanics

2 Dynamical Methods⇒ Time-Dependent SchrödingerEquation♦ Classical Molecular Dynamics♦ Semi-classical and Ab-Initio Molecular Dynamics


Ab Initio Methods

Ab Initio meaning "from first principles" methods solve the Schrödingerequation and does not rely on empirical or experimental data.

Begining with fundamental and physical properties, calculate howelectrons and nuclei interact.

The Schrödinger equation can be solved exactly only for a few systems

♦ Particle in a Box♦ Rigid Rotor♦ Harmonic Oscillator♦ Hydrogen Atom

For complex systems, Ab Initio methods make assumptions to obtainapproximate solutions to the Schrödinger equations and solve itnumerically.

"Computational Cost" of calculations increases with the accuracy of thecalculation and size of the system.


Ab Initio Methods

What can we predict with Ab Initio methods?Molecular Geometry: Equilibrium and Transition State

Dipole and Quadrupole Moments and polarizabilities

Thermochemical data like Free Energy, Energy of reaction.

Potential Energy surfaces, Barrier heights

Reaction Rates and cross sections

Ionization potentials (photoelectron and X-ray spectra) and Electronaffinities

Frank-Condon factors (transition probabilities, vibronic intensities)

Vibrational Frequencies, IR and Raman Spectra and Intensities

Rotational spectra

NMR Spectra

Electronic excitations and UV-VIS spectra

Electron density maps and population analyses

Thermodynamic quantities like partition function


Ab Initio Methods

Ab Initio TheoryBorn-Oppenheimer Approximation: Nuclei are heavier than electronsand can be considered stationary with respect to electrons. Also knowas "clamped nuclei" approximations and leads to idea of potentialsurface

Slater Determinants: Expand the many electron wave function in termsof Slater determinants.

Basis Sets: Represent Slater determinants by molecular orbitals, whichare linear combination of atomic-like-orbital functions i.e. basis sets


Born-Oppenheimer Approximation I

Solve time-independent Schrödinger equation

HΨ = EΨ

For many electron system:

H = −~2

2

∑α

∇2α

Mα︸︷︷︸Tn

− ~2

2me

∑i

∇2i︸︷︷︸

Te

+∑α>β

e2ZαZβ4πε0Rαβ︸︷︷︸

Vnn

−∑α,i

e2Zα4πε0Rαi︸︷︷︸

Ven

+∑i>j

e2

4πε0rij︸︷︷︸Vee︸︷︷︸

V

The wave function Ψ(R, r) of the many electron molecule is a function ofnuclear (R) and electronic (r) coordinates.

Motion of nuclei and electrons are coupled.

However, since nuclei are much heavier than electrons, the nucleiappear fixed or stationary.


Born-Oppenheimer Approximation II

Born-Oppenheimer Approximation: Separate electronic and nuclearmotion:

Ψ(R, r) = ψe(r; R)ψn(R)

Solve electronic part of Schrödinger equation

Heψe(r; R) = Eeψe(r; R)

BO approximation leads tothe concept of potentialenergy surface

V(R) = Ee + Vnn

−200

−100

0

100

200

300

1 2 3 4 5V

(kca

l/m

ol)

RHO−H(a0)

De

Re


Potential Energy Surfaces

The potential energy surface (PES) is multi-dimensional (3N − 6 fornon-linear molecule and 3N − 5 for linear molecule)The PES contains multiple minima and maxima.Geometry optimization search aims to find the global minimum of thepotential surface.Transition state or saddle point search aims to find the maximum of thispotential surface, usually along the reaction coordinate of interest.


Wavefunction Methods I

The electronic Hamiltonian (in atomic units, ~,me, 4πε0, e = 1) to besolved is

He = −12

∑i

∇2i −

∑α,i

ZαRiα

+∑i>j

1rij

+∑α>β

ZαZβRαβ

Calculate electronic wave function and energy

Ee =〈ψe | He | ψe〉〈ψe | ψe〉

The total electronic wave function is written as a Slater Determinant ofthe one electron functions, i.e. molecular orbitals, MO’s

ψe =1√N!

∣∣∣∣∣∣∣∣φ1(1) φ2(1) · · · φN(1)φ1(2) φ2(2) · · · φN(2)· · · · · · · · · · · ·φ1(N) φ2(N) · · · φN(N)

∣∣∣∣∣∣∣∣


Wavefunction Methods II

MO’s are written as a linear combination of one electron atomicfunctions or atomic orbitals (AO’s)

φi =N∑µ=1

cµiχµ

cµi ⇒ MO coefficientsχµ ⇒ atomic basis functions.

Obtain coefficients by minimizing the energy via Variational Theorem.

Variational Theorem: Expectation value of the energy of a trialwavefunction is always greater than or equal to the true energy

Ee = 〈ψe | He | ψe〉 ≥ ε0

Increasing N ⇒ Higher quality of wavefunction⇒ Higher computationalcost


Ab Initio Methods

The most popular classes of ab initio electronic structuremethods:

Hartree-Fock methods

♦ Hartree-Fock (HF)� Restricted Hartree-Fock (RHF): singlets� Unrestricted Hartree-Fock (UHF): higher multiplicities� Restricted open-shell Hartree-Fock (ROHF)

Post Hartree-Fock methods

♦ Møller-Plesset perturbation theory (MPn)♦ Configuration interaction (CI)♦ Coupled cluster (CC)

Multi-reference methods

♦ Multi-configurational self-consistent field (MCSCF)♦ Multi-reference configuration interaction (MRCI)♦ n-electron valence state perturbation theory (NEVPT)♦ Complete active space perturbation theory (CASPTn)


Hartree-Fock

1 Wavefunction is written as a single determinant

Ψ = det(φ1, φ2, · · ·φN)

2 The electronic Hamiltonian can be written as

H =∑

i

h(i) +∑i>j

v(i, j)

where h(i) = −12∇2

i −∑i,α

Zαriα

and v(i, j) =1rij

3 The electronic energy of the system is given by:

E = 〈Ψ|H|Ψ〉

4 The resulting HF equations from minimization of energy by applying ofvariational theorem:

f (x1)χi(x1) = εiχi(x1)

where εi is the energy of orbital χi and the Fock operator f , is defined as

f (x1) = h(x1) +∑

j

[Jj(x1)− Kj(x1)

]High Performance Computing @ Louisiana State University - http://www.hpc.lsu.edu March 25, 2011 15/66

Hartree-Fock

1 Jj ⇒ Coulomb operator⇒ average potential at x due to chargedistribution from electron in orbital χi defined as

Jj(x1)χi(x1) =

[∫χ∗j (x2)χj(x2)

r12dx2

]χi(x1)

2 Kj ⇒ Exchange operator⇒ Energy associated with exchange ofelectrons⇒ No classical interpretation for this term.

Kj(x1)χi(x1) =

[∫χ∗j (x2)χi(x2)

r12dx2

]χj(x1)

3 The Hartree-Fock equation are solved numerically or in a spacespanned by a set of basis functions (Hartree-Fock-Roothan equations)

χi =K∑µ=1

Cµiχµ∑ν

FµνCνi = εi

∑ν

SµνCνi

FC = SCε

Sµν =

∫dx1χ

∗µ(x1)χν(x1)

Fµν =

∫dx1χ

∗µ(x1)f (x1)χν(x1)


Hartree-Fock

1 The Hartree-Fock-Roothan equation is a pseudo-eigenvalue equation2 C’s are the expansion coefficients for each orbital expressed as a linear

combination of the basis function.3 Note: C depends on F which depends on C⇒ need to solve self-consistently.4 Starting with an initial guess orbitals, the HF equations are solved iteratively or

self consistently (Hence HF procedure is also known as self-consistent field orSCF approach) obtaining the best possible orbitals that minimize the energy.

SCF procedure

1 Specify molecule, basis functions and electronic state of interest2 Form overlap matrix S3 Guess initial MO coefficients C4 Form Fock Matrix F5 Solve FC = SCε

6 Use new MO coefficients C to build new Fock Matrix F7 Repeat steps 5 and 6 until C no longer changes from one iteration to the next.


What are Post Hartree-Fock Methods

1 In Hartree-Fock theory, electron motions of independent of each other i.e.uncorrelated.

2 However, this is not true. For two electrons with same spin| Ψ1(r1)α(ω1)Ψ2(r2)α(ω2)〉, the probability of finding electron 1 at r1 and electron2 at r2

P(r1, r2)dr1dr2 =12

(|Ψ1(r1)|2|Ψ2(r2)|2 + |Ψ1(r2)|2|Ψ2(r1)|2

− [Ψ∗1 (r1)Ψ2(r1)Ψ∗2 (r2)Ψ1(r2)

+ Ψ∗2 (r1)Ψ1(r1)Ψ∗1 (r2)Ψ2(r2)]) dr1dr2

Now P(r1, r1) = 0⇒ No two electrons with same spins can be at the same place⇒ "Fermi hole"

3 Same-spin electrons are correlated while different spin electrons are not.4 Energy difference between HF energy and the true energy is the correlation

energy

Ecorr = E0 − EHF


♦ Methods that improve the Hartree-Fock results by accounting for the correlation energyare known as Post Hartree-Fock methods

♦ The starting point for most Post HF methods is the Slater Determinant obtain fromHartree-Fock Methods.

♦ Configuration Interaction (CI) methods: Express the wavefunction as a linearcombination of Slater Determinants with the coeffcients obtained variationally

|Ψ〉 =∑

I

cI |ΨI〉

♦ Many Body Perturbation Theory: Treat the HF determinant as the zeroth order solutionwith the correlation energy as a pertubation to the HF equation.

H = H0 + λH′; εi = E(0)i + λE(1)

i + λ2E(2)i + · · ·

|Ψi〉 = |Ψ(0)i 〉+ λ|Ψ(1)

i 〉+ λ2|Ψ(2)i 〉 · · ·

♦ Coupled Cluster Theory:The wavefunction is written as an exponential ansatz

|Ψ〉 = eT |Ψ0〉

where |Ψ0〉 is a Slater determinant obtained from HF calculations and T is an excitationoperator which when acting on |Ψ0〉 produces a linear combination of excited Slaterdeterminants.


Scaling

Scaling Behavior Method(s)

N4 HF

N5 MP2

N6 MP3, CISD, CCSD, QCISD

N7 MP4, CCSD(T), QCISD(T)

N8 MP5, CISDT, CCSDT

N9 MP6

N10 MP7, CISDTQ, CCSDTQ


Density Functional Theory I

Density Functional Theory (DFT) is an alternative to wavefunction basedelectronic structure methods of many-body systems such as Hartree-Fock andPost Hartree-Fock.

In DFT, the ground state energy is expressed in terms of the total electrondensity.

ρ0(r) = 〈Ψ0|ρ|Ψ0〉

We again start with Born-Oppenheimer approximation and write the electronicHamiltonian as

H = F + Vext

where F is the sum of the kinetic energy of electrons and the electron-electroninteraction and Vext is some external potential.


Density Functional Theory II

Modern DFT methods result from the Hohenberg-Kohn theorem1 The external potential Vext, and hence total energy is a unique functional of

the electron density ρ(r)

Energy =〈Ψ | H | Ψ〉〈Ψ | Ψ〉 ≡ E[ρ]

2 The ground state energy can be obtained variationally, the density thatminimizes the total energy is the exact ground state density

E[ρ] > E[ρ0], ifρ 6= ρ0

If density is known, then the total energy is:

E[ρ] = T[ρ] + Vne[ρ] + J[ρ] + Enn + Exc[ρ]

where

Enn[ρ] =∑A>B

ZAZB

RABVne[ρ] =

∫ρ(r)Vext(r)dr

J[ρ] =12

∫ρ(r1)ρ(r2)

r12dr1dr2


Density Functional Theory III

If the density is known, the two unknowns in the energy expression are thekinetic energy functional T[ρ] and the exchange-correlation functional Exc[ρ]

To calculate T[ρ], Kohn and Sham introduced the concept of Kohn-Sham orbitalswhich are eigenvectors of the Kohn-Sham equation(

−12∇2 + veff(r)

)φi(r) = εiφi(r)

Here, εi is the orbital energy of the corresponding Kohn-Sham orbital, φi, and thedensity for an ”N”-particle system is

ρ(r) =

N∑i

|φi(r)|2

The total energy of a system is

E[ρ] = Ts[ρ] +

∫dr vext(r)ρ(r) + VH[ρ] + Exc[ρ]


Density Functional Theory IV

Ts is the Kohn-Sham kinetic energy which is expressed in terms of the Kohn-Sham orbitalsas

Ts[ρ] =N∑

i=1

∫dr φ∗i (r)

(−

12∇2)φi(r)

vext is the external potential acting on the interacting system (at minimum, for a molecularsystem, the electron-nuclei interaction), VH is the Hartree (or Coulomb) energy,

VH =12

∫drdr′

ρ(r)ρ(r′)|r − r′|

and Exc is the exchange-correlation energy.

The Kohn-Sham equations are found by varying the total energy expression with respectto a set of orbitals to yield the Kohn-Sham potential as

veff(r) = vext(r) +

∫ρ(r′)|r − r′|

dr′ +δExc[ρ]

δρ(r)

where the last term vxc(r) ≡δExc[ρ]

δρ(r)is the exchange-correlation potential.


Density Functional Theory V

The exchange-correlation potential, and the corresponding energy expression,are the only unknowns in the Kohn-Sham approach to density functional theory.

There are many ways to approximate this functional Exc, generally divided intotwo separate terms

Exc[ρ] = Ex[ρ] + Ec[ρ]

where the first term is the exchange functional while the second term is thecorrelation functional.

Quite a few research groups have developed the exchange and correlationfunctionals which are fit to empirical data or data from explicity correlatedmethods.

Popular DFT functionals (according to a recent poll)

♦ PBE0 (PBEPBE), B3LYP, PBE, BP86, M06-2X, B2PLYP, B3PW91, B97-D,M06-L, CAM-B3LYP

� http://www.marcelswart.eu/dft-poll/index.html

� http://www.ccl.net/cgi-bin/ccl/message-new?2011+02+16+009


http://www.marcelswart.eu/dft-poll/index.html

http://www.ccl.net/cgi-bin/ccl/message-new?2011+02+16+009

Semi-empirical Methods

Semi-empirical quantum methods:♦ Represents a middle road between the mostly qualitative results

from molecular mechanics and the highly computationallydemanding quantitative results from ab initio methods.

♦ Address limitations of the Hartree-Fock claculations, such asspeed and low accuracy, by omitting or parametrizing certainintegrals

integrals are either determined directly from experimental data orcalculated from analytical formula with ab initio methods or fromsuitable parametric expressions.

Integral approximations:♦ Complete Neglect of Differential Overlap (CNDO)♦ Intermediate Neglect of Differential Overlap (INDO)♦ Neglect of Diatomic Differential Overlap (NDDO) ( Used by PM3,

AM1, ...)

Semi-empirical methods are fast, very accurate when applied to moleculesthat are similar to those used for parametrization and are applicable to verylarge molecular systems.


Heirarchy of Methods

Hartree-‐Fock HF-‐SCF

Excita1on Hierarchy CIS,CISD,CISDT CCS,CCSD,CCSDT

Mul1configura1onal HF MCSCF,CASSCF

Perturba1on Hierarchy MP2,MP3,MP4

Mul1reference Perturba1on CASPT2, CASPT3

Full CI

Semi-‐empirical methods CNDO,INDO,AM1,PM3


Basis Sets I

Slater type orbital (STO) or Gaussian type orbital (GTO) to describe theAO’s

χSTO(r) = xlymzne−ζr

χGTO(r) = xlymzne−ξr2

where L = l + m + n is the total angular momentun and ζ, ξ are orbitalexponents.

0

0.5

1

0 1 2 3 4 5

Am

pli

tud

e

r (a0)

STOGTO


Basis Sets II

Why STOCorrect cups at r → 0

Desired decay at r →∞Correctly mimics H orbitals

Natural Choice for orbitals

Computationally expensive tocompute integrals andderivatives.

Why GTOWrong behavior at r → 0 andr →∞Gaussian × Gaussian =Gaussian

Analytical solutions for mostintegrals and derivatives.

Computationally less expensivethan STO’s


Basis Sets III

Pople family basis set1 Minimal Basis: STO-nG

♦ Each atom optimized STO is fit with n GTO’s♦ Minimum number of AO’s needed

2 Split Valence Basis: 3-21G,4-31G, 6-31G

♦ Contracted GTO’s optimized per atom.♦ Valence AO’s represented by 2 contracted GTO’s

3 Polarization: Add AO’s with higher angular momentum (L)

♦ 3-21G* or 3-21G(d),6-31G* or 6-31G(d),6-31G** or6-31G(d,p)

4 Diffuse function: Add AO with very small exponents for systems withdiffuse electron densities

♦ 6-31+G*, 6-311++G(d,p)


Basis Sets IV

Correlation consistent basis set♦ Family of basis sets of increasing sizes.

♦ Can be used to extrapolate basis set limit.

♦ cc-pVDZ: Double Zeta(DZ) with d’s on heavy atoms, p’s on H

♦ cc-pVTZ: triple split valence with 2 sets of d’s and 1 set of f’s on heavyatom, 2 sets of p’s and 1 set of d’s on H

♦ cc-pVQZ, cc-pV5Z, cc-pV6Z

♦ can be augmented with diffuse functions: aug-cc-pVXZ (X=D,T,Q,5,6)


Basis Sets V

Pseudopotentials or Effective Core Potentials♦ All Electron calculations are prohibitively expensive.

♦ Only valence electrons take part in bonding interaction leaving coreelectrons unaffected.

♦ Effective Core Potentials (ECP) a.k.a Pseudopotentials describeinteractions between the core and valence electrons.

♦ Only valence electrons explicitly described using basis sets.

♦ Pseudopotentials commonly used

� Los Alamos National Laboratory: LanL1MB and LanL2DZ� Stuttgard Dresden Pseudopotentials: SDDAll can be used.� Stevens/Basch/Krauss ECP’s: CEP-4G,CEP-31G,CEP-121G

♦ Pseudopotential basis are "ALWAYS" read in pairs

� Basis set for valence electrons� Parameters for core electrons


Molecular Mechanics

The potential energy of all systems in molecular mechanics iscalculated using force fields.

Molecular mechanics can be used to study small molecules as well aslarge biological systems or material assemblies with many thousands tomillions of atoms.

All-atomistic molecular mechanics methods have the followingproperties:

♦ Each atom is simulated as a single particle♦ Each particle is assigned a radius (typically the van der Waals

radius), polarizability, and a constant net charge (generally derivedfrom quantum calculations and/or experiment)

♦ Bonded interactions are treated as "springs" with an equilibriumdistance equal to the experimental or calculated bond length

The exact functional form of the potential function, or force field,depends on the particular simulation program being used.


Molecular Mechanics

General form of MolecularMechanics equations

E = Ebond + Eangle + Edihedral + EvdW + Eelec

=12

∑bonds

Kb(b− b0)2

+12

∑angles

Kθ(θ − θ0)2

+12

∑dihedrals

Kφ [1 + cos(nφ)]2

+∑

non−bonds

[(σ

r

)12−(σ

r

)6]

+q1q2

Dr


QM/MM

Quantum Mechanics(QM): Accurate, expensive (O(N4)), suitable forsmall systems.

Molecular Mechanics(MM): Approximate, does not treat electronsexplicitly, suitable for large systems such as enzymes and proteins,cannot simulate bon breaking/forming

What do we do if we want simulate chemical reaction in large systems?

Methods that combine QM and MM are the solution.

Such methods are called Hybrid QM/MM methods.

The basic idea is to partition the system into two (or more) parts1 The region of chemical interest is treated using accurate QM

methods eg. active site of an enzyme.2 The rest of the system is treated using MM or less accurate QM

methods such as semi-empirical methods or a combination of thetwo.

HTotal = HQM + HMM + HintQM−MM


QM/MM

ONIOM: divide the system into a real (full) system and the modelsystem. Treat the model system at high and low level. The total energyof the system is given by

E = E(low, real) + E(high,model)− E(low,model)

Empirical Valence Bond: Treat any point on a reaction surface as acombination of two or more valence bond structures

Effective Fragment Potential: add fragments to a standard QMtreatment, which are fully polarizable and are parameterized fromseparate ab initio calculations.


Molecular Dynamics

Why Molecular Dynamics?

Electronic Structure Methods are applicable to systems in gas phase under lowpressure (vaccum).

Majority of chemical reactions take place in solution at some temperature withbiological reactions usually at specific pH’s.

Calculating molecular properties taking into account such environmental effectswhich can be dynamical in nature are not adequately described by electronicstructure methods.

Molecular Dynamics

Generate a series of time-correlated points in phase-space (a trajectory).

Propagate the initial conditions, position and velocities in accordance withNewtonian Mechanics. F = ma = −∇V

Fundamental Basis is the Ergodic Hypothesis: the average obtained byfollowing a small number of particles over a long time is equivalent to averagingover a large number of particles for a short time.


Molecular Dynamics Theory

Solve the time-dependent Schrödinger equation

ı~ ∂∂t

Ψ(R, r, t) = HΨ(R, r, t)

with

Ψ(R, r, t) = χ(R, t)Φ(r, t)

and

H = −∑

I

~2

2MI∇2

I +−~2

2me∇2

i + Vn−e(r,R)︸︷︷︸He(r,R)

Obtain coupled equations of motion for electrons and nuclei:Time-Dependent Self-Consistent Field (TD-SCF) approach.

ı~∂Φ

∂t=

[−∑

i

~2

2me∇2

i + 〈χ|Vn−e|χ〉

]Φ

ı~∂χ∂t

=

[−∑

I

~2

2MI∇2

I + 〈Φ|He|Φ〉

]χ


Define nuclear wavefunction as

χ(R, t) = A(R, t) exp [iS(R, t)/~]

where A and S are real.Solve the time-dependent equation for nuclear wavefunction and takeclassical limit (~→ 0) to obtain

∂S∂t

+∑

I

~2

2MI(∇IS)2 + 〈Φ|He|Φ〉 = 0

an equation that is isomorphic with the Hamilton-Jacobi equation withthe classical Hamilton function given by

H({RI}, {PI}) =∑

I

~2

2MIP2

I + V({RI})

where

PI ≡ ∇IS and V({RI}) = 〈Φ|He|Φ〉

Obtain equations of nuclear motion from Hamilton’s equationdPI

dt= − dH

dRI⇒ MRI = −∇IV

dRI

dt=

dHdPI


Replace nuclear wavefunction by delta functions centered on nuclearposition to obtain

i~∂Φ

∂t= He(r, {RI})Φ(r; {RI}, t)

This approach of simultaneously solving the electronic and nucleardegrees of freedom by incorporating feedback in both directions isknown as Ehrenfest Molecular Dynamics.

Expand Φ in terms of many electron wavefunctions or determinants

Φ(r; {RI}, t) =∑

i

ci(t)Φi(r; {RI})

with matrix elements

Hij = 〈Φi|He|Φj〉

Inserting Φ in the TDSE above, we get

ı~ci(t) = ci(t)Hii − ı~∑

I,i

RIdijI

with non-adiabtic coupling elements given by

dijI (RI) = 〈Φi|∇I |Φj〉


Upto this point, no restriction on the nature of Φi i.e. adiabatic ordiabatic basis has been made.

Ehrenfest method rigorously includes non-adiabtic transitions betweenelectronic states within the framework of classical nuclear motion andmean field (TD-SCF) approximation to the electronic structure.

Now suppose, we define {Φi} to be the adiabatic basis obtained fromsolving the time-independent Schrödinger equation,

He(r, {RI})Φi(r; {RI}) = Ei({RI})Φi(r; {RI})

The classical nuclei now move along the adiabatic orBorn-Oppenheimer potential surface. Such dynamics are commonlyknown as Born-Oppenheimer Molecular Dynamics or BOMD.

If we restrict the dynamics to only the ground electronic state, then weobtain ground state BOMD.

If the Ehrenfest potential V({RI}) is approximated to a global potentialsurface in terms of many-body contributions {vn}.

V({RI}) ≈ Vapproxe (R) =

N∑I=1

v1(RI) +

N∑I>J

v2(RI ,RJ) +

N∑I>J>K

v3(RI ,RJ ,RK) + · · ·


Thus the problem is reduced to purely classical mechanics once the{vn} are determined usually Molecular Mechanics Force Fields. Thisclass of dynamics is most commonly known as Classical MolecularDynamics.If we follow Lagrangian mechanics and write down the Lagrangian forthe system of interest.

L = T − V

Corresponding Newton’s equation of motion are then obtained from theassociated Euler-Lagrange equations,

ddt∂L∂RI

=∂L∂RI

The Lagrangian for ground state BOMD is

LBOMD =∑

I

12

MIR2I −min

Φ0〈Φ0|He|Φ0〉

and equations of motions

MIRI =ddt∂LBOMD

∂RI=∂LBOMD

∂RI= −∇I min

Φ0〈Φ0|He|Φ0〉


Extended Lagrangian Molecular Dynamics (ELMD): Extend theLagrangian by adding kinetic energy of fictitious particles, i.e. treatquantum degrees of freedom as classical particles and obtain theirequation of motions from Euler-Lagrange equations.Car-Parrinello Molecular Dynamics (CPMD):

LCPMD =∑

I

12

MIR2I +

∑i

12µi〈φi|φi〉 − 〈Φ0|He|Φ0〉+ constraints

Atom centered Density Matrix Propagation (ADMP):

LADMP =12

Tr(VT MV) +12µTr(WW)− E(R,P)− Tr[Λ(PP− P)]

whereddt

P = W

curvy-steps ELMD (csELMD):

LcsELMD =∑

I

12

MIR2I +

12µ∑i<j

∆ij − Tr[(h +12

H)P]− Vnuc

where

P(λ) = eλ∆P(0)e−λ∆


Molecular Dynamics: Methods and Programs

Electronic energy obtained fromMolecular Mechanics⇒ Classical Molecular Dynamics

1 LAMMPS2 NAMD3 Amber4 Gromacs

Ab-Initio Methods⇒ Quantum or Ab-Initio MolecularDynamics

1 Born-Oppenheimer Molecular Dynamics: Gaussian,GAMESS

2 Extended Lagrangian Molecular Dynamics: VASP, CPMD,Gaussian (ADMP), NWCHEM(CPMD), QChem (curvy-stepsELMD)

3 Time Dependent Hartree-Fock and Time Dependent DensityFunctional Theory: Gaussian, GAMESS, NWCHEM, QChem

4 Multiconfiguration Time Dependent Hartree(-Fock),MCTDH(F)

5 Non-Adiabatic and Ehrenfest Molecular Dynamics, MultipleSpawning, Trajectory Surface Hopping

6 Wavepacket Methods: Gaussian (QWAIMD)


Classical Molecular DynamicsAdvantages

1 Large Biological Systems2 Long time dynamics

Disadvantages1 Cannot describe Quantum Nuclear Effects

Ab Initio and Quantum DynamicsAdvantages

1 Quantum Nuclear EffectsDisadvantages

1 ∼ 100 atoms2 Full Quantum Dynamics ie treating nuclei quantum

mechanically: less than 10 atoms3 Picosecond dynamics at best


Software QB Eric Louie Oliver Painter Poseidon Philip Tezpur

Amber X X X X X X X XDesmond XDL_Poly X X X X X X X XGromacs X X X X X X X XLAMMPS X X X X X X X X

NAMD X X X X X X XOpenEye X X X X X X X X

CPMD X X X X X X XGAMESS X X X X X X X XGaussian X X X X X XNWCHEM X X X X X X XPiny_MD X X X X X X X X

Software Bluedawg Ducky Lacumba Neptune Zeke Pelican

Amber X X XGromacs X X X X X XLAMMPS X X X X X X

NAMD X X X X XCPMD X X X X X

Gaussian X X X X XNWCHEM X X X X X XPiny_MD X X X X X X


Computational Chemistry Programs

Commercial Software: Q-Chem, Jaguar,CHARMMGPL/Free Software: ACES, ABINIT, Octopushttp://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs

http://www.ccl.net/chemistry/links/software/index.shtml

http://www.redbrick.dcu.ie/~noel/linux4chemistry/


http://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs

http://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs





Job Types and Keywords I

Job Type Gaussian GAMESS NWCHEM# keyword runtyp= task

Energy sp energy energyForce force gradient gradient

Geometry optimization opt optimize optimizeTransition State opt=ts sadpoint saddle

Frequency freq hessian∗ frequencies, freqPotential Energy Scan scan surface X

Excited State X X XReaction path following irc irc X

Molecular Dynamics admp, bomd drc dynamics, Car-ParrinelloPopulation Analysis pop pop X

Electrostatic Properties prop X XMolecular Mechanics X X X

Solvation Models X X X

X=⇒ method exists and keyword requires more than one options


Job Types and Keywords II

Population Analysis in NWCHEM requires a PROPERTY · · · END block with variousoptions for different properties

propertydipolemulliken

end

Frequency calculations in GAMESS at end of optimization is carried out by addingHSSEND=.T. keyword in the STATPT control line

$STATPT HSSEND=.T. $END

Excited State Calculations include TDHF, TDDFT, CIS, CC methods

QM/MM Methods

Gaussian: ONIOM

GAMESS: Effective Fragment Potential, $EFRAG block

NWCHEM: task qmmm


Job Types and Keywords III

Dynamics Calculations:

Gaussian:BOMD: Born-Oppenheimer Molecular DynamicsADMP: Atom centered Density Matrix Propagation (anextended Lagrangian Molecular Dynamics similar to CPMD)and ground state BOMD

GAMESS:DRC: Direct Dynamics, a classical trajectory method basedon "on-the-fly" ab-initio or semi-empirical potential energysurfaces

NWCHEM:Car-Parrinello: Car Parrinello Molecular Dynamics (CPMD)DIRDYVTST: Direct Dynamics Calculations usingPOLYRATE with electronic structure from NWCHEM


Related HPC Tutorials

Fall Semester

Introduction to Gaussian/Electronic Structure Methods

Spring Semester

MD: Programming to ProductionApril 6th

Introduction to CPMD/Ab Initio Molecular DynamicsApril 27th


Useful LinksAmber:http://ambermd.org

Desmond:http://www.deshawresearch.com/resources_desmond.html

DL_POLY:http://www.cse.scitech.ac.uk/ccg/software/DL_POLY

Gromacs:http://www.gromacs.org

LAMMPS:http://lammps.sandia.gov

NAMD:http://www.ks.uiuc.edu/Research/namd

CPMD: http://www.cpmd.org

GAMESS: http://www.msg.chem.iastate.edu/gamess

Gaussian: http://www.gaussian.com

NWCHEM: http://www.nwchem-sw.org

PINY_MD:http://homepages.nyu.edu/~mt33/PINY_MD/PINY.html

Basis Set: https://bse.pnl.gov/bse/portal


http://ambermd.org

http://www.deshawresearch.com/resources_desmond.html

http://www.deshawresearch.com/resources_desmond.html

http://www.cse.scitech.ac.uk/ccg/software/DL_POLY

http://www.cse.scitech.ac.uk/ccg/software/DL_POLY

http://www.gromacs.org

http://lammps.sandia.gov

http://www.ks.uiuc.edu/Research/namd

http://www.cpmd.org

http://www.msg.chem.iastate.edu/gamess

http://www.gaussian.com

http://www.nwchem-sw.org

http://homepages.nyu.edu/~mt33/PINY_MD/PINY.html

http://homepages.nyu.edu/~mt33/PINY_MD/PINY.html

https://bse.pnl.gov/bse/portal

Further ReadingDavid Sherill’s Notes at Ga Tech:http://vergil.chemistry.gatech.edu/notes/index.html

Mark Tuckerman’s Notes at NYU: http://www.nyu.edu/classes/tuckerman/quant.mech/index.html

Modern Quantum Chemistry: Introduction to Advanced ElectronicStructure Theory, A. Szabo and N. Ostlund

Introduction to Computational Chemistry, F. Jensen

Essentials of Cmputational Chemistry - Theories and Models, C. J.Cramer

Exploring Chemistry with Electronic Structure Methods, J. B. Foresmanand A. Frisch

Molecular Modeling - Principles and Applications, A. R. Leach

Computer Simulation of Liquids, M. P. Allen and D. J. Tildesley

♦ Modern Electronic Structure Theory, T. Helgaker, P. Jorgensen and J.Olsen (Highly advanced text, second quantization approach toelectronic structure theory)


http://vergil.chemistry.gatech.edu/notes/index.html

http://www.nyu.edu/classes/tuckerman/quant.mech/index.html

http://www.nyu.edu/classes/tuckerman/quant.mech/index.html

On LONI Linux Systems

/home/apacheco/CompChem

AIMD

CMD

ElecStr

ADMP

BOMD

CPMD

DRC

Amber

Desmond

Gromacs

LAMMPS

NAMD

OptFreq

PES

ExState

QMMM


Using Gaussian on LONI Systems

Site specific license1 Gaussian 03 and 09

LSU Users: EricLatech Users: Painter, Bluedawg

2 Gaussian 03ULL Users: Oliver, ZekeTulane Users: Louie, DuckySouthern Users: Lacumba

3 UNO Users: No License

Add +gaussian-03/+gaussian-09 to your .soft file and resoftIf your institution has license to both G03 and G09, haveonly one active at a given time.


Example Job submission script on Intel x86#!/bin/tcsh#PBS -A your_allocation# specify the allocation. Change it to your allocation#PBS -q checkpt# the queue to be used.#PBS -l nodes=1:ppn=4# Number of nodes and processors#PBS -l walltime=1:00:00# requested Wall-clock time.#PBS -o g03_output# name of the standard out file to be "output-file".#PBS -j oe# standard error output merge to the standard output file.#PBS -N g03test# name of the job (that will appear on executing the qstat command).

# setup g03 variablessource $g03root/g03/bsd/g03.loginset NPROCS=‘wc -l $PBS_NODEFILE |gawk ’//{print $1}’‘setenv GAUSS_SCRDIR /scratch/$USER# cd to the directory with Your input filecd ~apacheco/g03test# Change this line to reflect your input file and output fileg03 < g03job.inp > g03job.out

Linda Accessset NODELIST = ( -vv -nodelist ’"’ ‘cat $PBS_NODEFILE‘ ’"’ -mp 4)setenv GAUSS_LFLAGS " $NODELIST "g03l < g03job.inp > g03job.out


Example Job submission script on P5#!/bin/tcsh# @ account_no = your_allocation# @ requirements = (Arch == "Power5")# @ environment = LL_JOB=TRUE ; MP_PULSE=1200# @ job_type = parallel# @ node_usage = shared# @ wall_clock_limit = 12:00:00# @ initialdir = /home/apacheco/g03test# @ class = checkpt# @ error = g03_$(jobid).err# @ queue

# setup g03 variablessource $g03root/g03/bsd/g03.login# setup and create Gaussian scratch directorysetenv GAUSS_SCRDIR /scratch/default/$USERmkdir -p $GAUSS_SCRDIR# cd to the directory with Your input filecd ~apacheco/g03test# Change this line to reflect your input file and output fileg03 < g03job.inp > g03job.out


Sample Input%chk=h2o-opt-freq.chk%mem=512mb%NProcShared=4

#p b3lyp/6-31G opt freq

H2O OPT FREQ B3LYP

0 1OH 1 r1H 1 r1 2 a1

r1 1.05a1 104.5

Input Descriptioncheckpoint fileamount of memorynumber of smp processorsblank lineJob descriptionblank lineJob Titleblank lineCharge MultiplicityMolecule DescriptionZ-matrix formatwith variables

blank linevariable value

blank line


Using GAMESS on LONI Systems

Add +gamess-12Jan2009R1-intel-11.1 (on Queenbee) toyour .soft and resoft

Job submission script#!/bin/bash#PBS -A your_allocation#PBS -q checkpt#PBS -l nodes=1:ppn=4#PBS -l walltime=00:10:00#PBS -j oe#PBS -N gamess-exam1

export WORKDIR=$PBS_O_WORKDIRexport NPROCS=‘wc -l $PBS_NODEFILE | gawk ’//{print $1}’‘export SCRDIR=/work/$USER/scrif [ ! -e $SCRDIR ]; then mkdir -p $SCRDIR; firm -f $SCRDIR/*

cd $WORKDIRrungms h2o-opt-freq 01 $NPROCS h2o-opt-freq.out $SCRDIRcp -p $SCRDIR/$OUTPUT $WORKDIR/


Sample Input$CONTRL SCFTYP=RHF RUNTYP=OPTIMIZECOORD=ZMT NZVAR=0 $END

$STATPT OPTTOL=1.0E-5 HSSEND=.T. $END$BASIS GBASIS=N31 NGAUSS=6NDFUNC=1 NPFUNC=1 $END

$DATAH2O OPTCnv 2

OH 1 rOHH 1 rOH 2 aHOH

rOH=1.05aHOH=104.5$END

Input DescriptionJob control data

geometry search control6-31G** basis set

molecular data controlTitleSymmetry group and axis

molecule description inz-matrix

variables

end molecular data control


Using NWCHEM on LONI Systems

Add +nwchem-5.1.1-intel-11.1-mvapich-1.1 (onQueenbee) to your .soft and resoft

Job submission script#!/bin/sh##PBS -q checkpt#PBS -M [email protected]#PBS -l nodes=1:ppn=4#PBS -l walltime=0:30:00#PBS -V#PBS -o nwchem_h2o.out#PBS -e nwchem_h2o.err#PBS -N nwchem_h2o

export EXEC=nwchemexport EXEC_DIR=/usr/local/packages/nwchem-5.1-mvapich-1.0-intel-10.1/bin/LINUX64/export WORK_DIR=$PBS_O_WORKDIRexport NPROCS=‘wc -l $PBS_NODEFILE |gawk ’//{print $1}’‘

cd $WORK_DIRmpirun_rsh -machinefile $PBS_NODEFILE -np $NPROCS $EXEC_DIR/$EXEC \

$WORK_DIR/h2o-opt-freq.nw >& $WORK_DIR/h2o-opt-freq.nwo


Sample Inputtitle "H2O"echocharge 0geometryzmatrixOH 1 r1H 1 r1 2 a1variablesr1 1.05a1 104.5endendbasis noprint

* library 6-31GenddftXC b3lypmult 1

endtask dft optimizetask dft energytask dft freq

Input DescriptionJob titleecho contents of input filecharge of moleculegeometry description inz-matrix format

variables used with values

end z-matrix blockend geometry blockbasis description

dft calculation options

job type


Introduction to Computational Chemistrycct.lsu.edu/~apacheco/tutorials/Intro2CompChem.pdf ·...

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Transcript of Introduction to Computational Chemistrycct.lsu.edu/~apacheco/tutorials/Intro2CompChem.pdf ·...