Molecular Electric Moments Calculated By Using Natural Orbital Functional Theory · 2016. 5....

Molecular Electric MomentsCalculated By Using Natural Orbital

Functional Theory

Ion Mitxelena Echeverria

March 2016

DirectorProf. Dr. Mario Piris

RefereesProf. Dr. Xabier LopezProf. Dr. Jesus Ugalde

Contents

Abstract 4

1 Introduction 51.1 The NOF Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 RDMs and The Energy Functional . . . . . . . . . . . . . . . 51.1.2 Piris Natural Orbital Functional . . . . . . . . . . . . . . . . . 10

1.2 Molecular Electrostatic Moments . . . . . . . . . . . . . . . . . . . . 141.2.1 Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 Octupole Moment . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Methodology 21

3 Results and Discussion 233.1 Dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Quadrupole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Octupole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Conclusions 31

Acknowledgments 35

Bibliography 37

5 Appendix I 43

i

“The underlying physical laws necessary for the mathematical theoryof a large part of physics and the whole of chemistry are thus com-pletely known, and the difficulty is only that the exact application ofthese laws leads to equations much too complicated to be soluble. Ittherefore becomes desirable that approximate practical methods of ap-plying quantum mechanics should be developed, which can lead to anexplanation of the main features of complex atomic systems without toomuch computation.”

(P.A.M. Dirac, London, 1929)

1

Abstract

The interpretation and understanding of intermolecular forces, particularly thoserelating to long-range interactions, require knowledge of the electrostatic moments[3, 4]. In other words, the complete determination of the charge density and cor-responding response functions is computationally heavy and conceptually complex,hence the electric moments are essential to provide simple ways to figure out theelectric field behaviour of molecules. In principle, one can experimentally find thecomponents of the electric field at each point, but it turns into a formidable task forlarge molecular systems. There are several techniques to determine experimentallythe dipole moments [22, 48], but it is still very difficult to obtain precise experimen-tal values of higher multipole moments such as quadrupole or octupole moments[3, 14, 8], independently of the experimental conditions. Theoretical calculationsare therefore essential but challenging for quantum chemistry methods. In fact,the accurate calculation of these properties is highly dependent on the method em-ployed [19], either regarding approximate density functionals [7] or methods basedon wavefunctions [54, 34]. Consequently, calculating the multipole moments is away to assess any electronic structure method.

One-electron reduced density matrix functional theory (1-RDMFT) is a promisingapproach to the problem of electron correlation based on the existence of a func-tional of the 1st-order reduced density matrix (1-RDM) [9]. In this context, NaturalOrbital Functional Theory (NOFT) [59] has emerged in recent years [20, 56] asan alternative method to both density functional theory [55] and WFN methods[49]. Since the interactions between electrons are pairwise in the Hamiltonian ofa many-electron system, the corresponding energy can be exactly determined interms of the two-particle reduced density matrix (2-RDM), which is reconstructedfrom the 1-RDM in 1-RDMFT. The kinetic energy of the electrons and the attrac-tive Coulomb potential between the nuclei and the electrons, that together formthe core-Hamiltonian term, are exactly defined in terms of the 1-RDM, so onlythe electron-electron correlation term has to be approximated in 1-RDMFT. In thenatural orbital (NO) representation the 1-RDM is diagonal, thereby the energy func-tional is written in terms of the NOs and the corresponding diagonal terms, knownas the occupation numbers (ONs), so that 1-RDMFT reduces to NOFT. In the lastdecade Piris has proposed a particular reconstruction of the 2-RDM based on thecumulant expansion [38, 46, 58], thus, several approximations to the cumulant leadto different implementations, known in the literature as PNOFi (i=1-6) [60, 61].

In the present master thesis, we investigate the performance of the Piris Natural Or-

3

bital Functional 6 (PNOF6) on the calculation of molecular electrostatic moments.The dipole and quadrupole moments of the molecules H2, HF, BH, HCl, H2O, H2CO,C2H2, C2H4, C2H6, C6H6, CH3CCH, CH3F, HCCF, ClF, CO, CO2, O3, N2, NH3,and PH3 are computed and compared to experimental data, as well as with thetheoretically computed values of Bundgen et al. [54] who used the multi-referencesingle and double excitation configuration interaction (MRSD-CI) method, and thecoupled-cluster single and doubles (CCSD) values calculated by us. Moreover, theoctupole moment of CH4, a molecule without dipole and quadrupole moments isalso studied. All calculations are carried out at equilibrium experimental geome-tries, using the triple-ζ Gaussian basis set with polarization functions developed bySadlej [75, 76]. The agreement between PNOF6(Nc) results and experimental datais satisfactory, so PNOF6(Nc) is able to predict electric properties of highly corre-lated systems. To our knowledge, this is the first NOF study of higher multipolemoments such as quadrupole and octupole moments.

4

1 Introduction

In this section we briefly give the theoretical background that is relevant for thiswork. First, we introduce the Natural Orbital Functional Theory as well as the lastfunctional proposed by Piris, known in the literature as PNOF6. Afterwards, theprincipal molecular electrostatic moments are described, from their practical andfundamental interest to their mathematical definition and implementation.

1.1 The NOF Theory

In the last few years, the development and improvement in computer hardwareand software has allowed the simulation of large electronic systems. The most em-ployed electronic structure methods are those based on Density Functional Theory(DFT) [55] due to their favorable efficiency-cost ratio; nevertheless, approximatedensity functionals still present several drawbacks, e.g., the inability to describehighly degenerate systems [51], or the poor description of dispersion interactions.On the other hand, methods based on the N -particle wave function (WFN) [49]give the most accurate description of the electronic structure; however, they scaleunfavourably with the number of functions present in the basis set, for instance, theCoupled Cluster Singles Doubles and perturbative Triplet excitations (CCSD(T))method, often referred as “The Gold-Standard Method” in quantum chemistry,scales formally as O (N7). Therefore, the corresponding large computational costlimits their use to systems with a small number of electrons.Natural Orbital Functional Theory (NOFT) has emerged in recent years [20] as analternative method to both density functional theory and WFN methods. One ofthe possible strategies towards development of one-electron reduced density matrixfunctionals consists of reconstructing the 2-RDM by using the cumulant expansion[38, 46, 58], and approximating the cumulant part. Recently, Piris proposed a seriesof functionals based on this approximation, these are called PNOFi (i=1-6) [60, 61].

1.1.1 RDMs and The Energy Functional

In the context of the Born-Oppenheimer approximation, the Hamiltonian of a manyelectron system is the sum of one- and two-electron operators, hence the corre-sponding energy can be determined exactly from the knowledge of the 1st- and

5

Chapter 1 Introduction

2nd- order reduced density matrices (1- and 2-RDMs). Generally speaking, for all0 < p < N, p ∈ Z, the p−order reduced density matrix is an object simpler thanthe N -electron wave function, and is defined as

Γp(x′

1 · · · x′p|x1 · · · xp

)=

(Np

)ˆψ∗(x′1x′2 · · · x′p · · · xN

)ψ (x1x2 · · · xp · · · xN) dxp+1 · · · dxN

. (1.1)

where each coordinate xi is a combination of a space coordinate ri and a spincoordinate si. Note that in (1.1) the system is characterized by the normalizedwave function ψ. Let us expand Γ1

(x′

1|x1)and Γ2

(x′

1x′2|x1x2

)in the complete

orthonormal set of single-particle functions φi (x), also referred to as spin orbitals(SOs),

Γ1(x′

1|x1)

= ∑ik Γ1

kiφ∗k

(x′

1

)φi (x1)

Γ2(x′

1x′2|x1x2

)= ∑

ik Γ2kl,ijφ

∗k

(x′

1

)φ∗l(x′

2

)φi (x1)φj (x2)

. (1.2)

Recall that Lowdin’s normalization convention [41] has been used along this work,so the trace of the 1-RDM is equal to the number of electrons, Tr Γ = N , and thetrace of the 2-RDM equals the number of electron pairs, i.e Tr D =

(N2

)= N(N−1)

2 .In second quantization notation, the 1- and 2-RDMs, denoted hereafter by Γ and Drespectively, can be expressed as

Γji = ⟨ψ|a+j ai|ψ⟩, (1.3)

and

Dkl,ij = 12⟨ψ|a+

k a+l aj ai|ψ⟩. (1.4)

From the anticommutation relations of annihilation and creation operators is straight-forward to derive the properties of reduced density matrices. Just cite here the mostimportant properties such as Hermiticity,

Γji = Γ∗ij(Γ = Γ+

), Dkl,ij = D∗ij,kl (D = D∗) , (1.5)

6

1.1 The NOF Theory

symmetry and antisymmetry,

Dkl,ij = Dlk,ji, Dkl,ij = −Dlk,ij = −Dkl,ji, (1.6)

and positivity of diagonal elements,

Γii ≥ 0, Dij,ij ≥ 0. (1.7)

The last property is very interesting from the point of view of physical interpreta-tions. The Γii element represents the probability of finding a particle in an orbitalφi, while the other electrons occupy arbitrary spin orbitals. Similarly, Dij,ij is theprobability of finding one particle in an orbital φi and another in an orbital φj, whenall other particles occupy arbitrary orbitals. Thus, condition (1.7) arises from thefact that any probability must be positive or zero by definition.Also, the 2-RDM must fulfill the sum rule

∑p

Dpqrq = (N − 1)npδpr. (1.8)

So far, we have introduced the basic concepts and notation of RDMs, paying spe-cial attention to the one- and two-matrices. In fact, since the Hamiltonian of amany-electron system contains at most two-particle interactions, the exact elec-tronic energy of N -electron systems is given explicitly in terms of the one- andtwo-matrices

E =∑ik

Γkihik +∑ijkl

Dkl,ij⟨kl|ij⟩, (1.9)

where hik corresponds to the one-electron matrix elements of the core-Hamiltonian,given by the kinetic energy of the electrons and the attractive Coulomb potentialbetween the nuclei and the electrons,

hik =ˆdxφ∗k (x)

[−1

2∇2 −

∑I

ZI|r− rI |

]φi (x) , (1.10)

and ⟨kl|ij⟩ are the two-electron integrals corresponding to the repulsive Coulombpotential between the electrons,

⟨kl|ij⟩ =ˆdx1dx2φ

∗k (x1)φ∗l (x2) r−1

12 φi (x1)φj (x2) . (1.11)

7


Note that atomic units are used. It is worth pointing out that any RDM can becalculated from RDMs of higher order according to the contraction relation, whichis straightforward from the general definition (1.1). Thus, the energy functional(1.9) is an exact functional of the 2-RDM, since the 1-RDM can be expressed as afunction of D:

Γ(x′

1|x1)

= 2N − 1

ˆdxD

(x′

1x|x1x). (1.12)

Note that the whole 2-RDM is necessary to calculate the 1-RDM from the two-matrix according to (1.12), thus, though the electron-electron interaction term in(1.9) can be expressed as a function only of diagonal terms of the 2-RDM, sincethe 1-RDM is required to calculate the kinetic energy and nuclear attraction terms,then the diagonal elements of the 2-RDM are not sufficient to determine completelythe energy functional (1.9). Conversely, it is interesting to investigate how much theknowledge of a lower-order density matrix determines higher order reduced densitymatrices or even the wave function, for instance, in the case of Hartree-Fock the wavefunction of the system is entirely determined by the 1-RDM. This is the strategyadopted in Natural Orbital Functional Theory, where the 2-RDM is approximatelyreconstructed from the 1-RDM.The motivation of NOFT, and more generally of one-electron reduced density matrixfunctional theory (1-RDMFT), arises from the fact that, since the kinetic energy andnuclear attraction terms are entirely defined using Γ without any approximation, onecan build the energy functional for any N -electron system in terms of the 1-RDMapproximating only the electron-electron interaction term, usually called electroncorrelation term. In 1974, Gilbert [18] gave a proof for the existence of the 1-RDMenergy functional, analog to the Hohenberg-Kohn theorem [30]. Besides, Levy [39]proved that the 1-RDM functional does not require v-representability, so only N -representability is required to the existence of the functional. Thereby the externalpotential v, implicit in the core-Hamiltonian (1.10), may be nonlocal as well as localand that a spin-dependent Hamiltonian may be used. In summary, the one-matrixis the key variable in NOF theory, and it plays the role analogue to the N−particlewave function (ψ) in ab−initio approaches or the density (ρ) in DFT. The 1-RDM ismuch more simpler than the N−particle wave function, but it is more sophisticatedthan the electronic density, hence NOFT is expected to be more reliable and accuratethan DFT at the same time that the computational cost required by accurate ab−initio methods is reduced.Let us focus on NOFT, which is located in the framework of 1-RDMFT. Since the 1-RDM is Hermitian, there exists a unitary transformation of the spin-orbitals φi (x)which transforms this matrix to diagonal form, composed only by the eigenvaluesni. The corresponding orbitals are called Natural Orbitals (NOs) and the eigen-values of the diagonalized matrix are known as their Occupation Numbers (ONs).

8

1.1 The NOF Theory

It is important to note that functionals currently in use are only known in the basiswhere the 1-RDM is diagonal, so that they are not functionals explicitly dependenton the 1-RDM and retain some dependence on the 2-RDM. Actually, the pictureis invariant with respect to unitary transformations of the spin-orbitals φi (x)only in the special Hartree-Fock case, whereas in general the Lagrangian matrix Λ[70] and the 1-RDM do not conmute; [Λ,Γ] 6= 0, therefore, they cannot be simul-taneously brought to the diagonal form and thereby the orbitals change from onerepresentation to another [69].In any case, the spectral decomposition of Γ leads to the simple mathematical form

Γki = niδki, Γ (x′i|xi) =∑i

niφ∗i (x′i)φi (xi) . (1.13)

Recall that these orbitals are physically meaningful, because they are directly associ-ated with electron occupancies. According to expansion (1.13), the energy functional(1.9) is expressed in NOF theory by

E [N, ni , φi (x)] =∑i

nihik + Vee [N, ni , φi (x)] , (1.14)

where Vee [N, ni , φi (x)] represents the electron-electron repulsion energy func-tional. To solve the many electron problem it is necessary to obtain the minimumenergy that correspond to the ground state of the system. From equation (1.14),the energy minimization problem consists in searching the set of NOs and ONsthat minimize the energy functional (1.14), nevertheless, minimization of the energywithout imposing general and required constraints may lead to energies below theexact energy. Indeed, it is required that the corresponding 1-RDM comes from anormalized antisymmetric N -electron wave function [9], i.e

Γ(x′

1|x1)

= N⟨ψ (x′1x′2 · · ·N) |ψ (x1x2 · · ·N)⟩2...N , (1.15)

and thereby the search is constrained to those 1-RDM that fulfill this condition1.This problem is known as the N -representability problem of the one-matrix. Cole-man [9] proved that any 1-RDM, in its spectral decomposition, is N -representable,i.e fulfills relation (1.15), if

´Γ(x′ |x

)dx = N and all ONs satisfy the inequal-

ity 0 ≤ ni ≤ 1. In the limit, i.e having no fractional occupancies (ni = 0, 1), thecorresponding 1-RDM is idempotent (Γ2 = Γ), and is what we obtain in the Hartree-Fock case. Therefore, having fractional occupancies is essential in order to describe

1Equation (1.15) is a particular case of (1.1) corresponding to p = 1.

9


properly systems with high multiconfigurational character, normally associated tonon-dynamical correlation.Anyway, N -representability constraints for 1-RDMs are not sufficient to guaranteethat the reconstructed 2-RDM isN -representable, i.e theN -representability problemof the 2-RDM is reflected on the reconstruction functional in NOFT. In other words,there must be a one-to-one correspondence between the energy E[ψ] expressed as afunction of the N−particle wave function ψ and the energy E[D], written in termsof the 2-RDM, so that the functional N -representability problem is equivalent to theN -representability problem of the 2-matrix. Unfortunately, a tractable solution tothe N -representability problem has been not found yet, although several advanceshave been obtained in the last few years [72, 9, 17, 47]. Recently, Mazziotti [47] hasproposed the so called (2,q)-positivity conditions, being q the order of the higherq−RDM used as the starting point for the derivation of the condition. Recall thatin the case of the (2,2)- and (2,3)-positivity conditions these are reduced to the well-known D, Q G, T1, and T2 conditions [47, 12, 9], whereas when q equals the rankof the one-electron basis set, the positivity conditions are complete. In practice,taking together, the D, Q and G conditions [26] are quite restrictive regarding thevariational calculation of the 2-RDM, that is why Piris enforce them in order to getimproved reconstruction of the 2-RDM and thereby of the energy functional (1.14).This strategy to get approximate forms of the 2-RDM is known as the Bottom−Upmethod [66]. Truthfully, the G matrix (associated to condition G) has a blockof dimension RxR, where R represents the number of functions of the basis set,and due to this basis set dependence there is no analytical way in order to imposethis condition to the reconstruction functional, so that one cannot guarantee thefulfillment of condition G. Fore more details see [26, 9, 47].

1.1.2 Piris Natural Orbital Functional

In the last decade, a series of Natural Orbital Functionals has been proposed by Pirisand collaborators (PNOFi, i=1, 6) [60, 61]. In all cases the reconstruction functionalis based on the cumulant expansion [38, 46] of the two-electron matrix, and findingan approximation for the cumulant part. In detail, the 2-RDM is expressed as anantisymmetrized product of the 1-RDMs and a correction to it, namely

Dkl,ij = 12 (ΓkiΓlj − ΓliΓkj) + λkl,ij. (1.16)

Note that any explicit dependence of 2-RDM on the NOs is neglected because theenergy functional (1.14) has already a strong dependence on them via the one- andtwo-electron integrals. Since the first two terms in (1.16) correspond to the Hartree-Fock (HF) approximation, the cumulant matrix, usually called the pair correlationmatrix, must vanish for a single Slater determinant. The HF term satisfies relations

10

1.1 The NOF Theory

(1.5)-(1.7), so the cumulant matrix should satisfy these relations too. In principle,λkl,ij has a dependence of four indices but this implies a huge computational cost,therefore, it is approximated by two-index matrices ∆ and Π expressed in terms ofthe NOs and their ONs. Following the Bottom − Up method proposed by Piris [66]to generate improved reconstruction functionals, these matrices are built such thatthe (2,2)-positivity conditions are enforced to ensure the N -representability of theenergy functional (1.14). Since the set of molecules studied in this work is limitedto closed-shell systems, we will focus on the general expressions for singlet states.Thus, the corresponding spin structure for the cumulant matrix is

λσσpq,rt = −∆pq

2 (δprδqt − δptδqr) , σ = α, β

λαβpq,rt = −∆pq

2 δprδqt + Πpr2 δpqδrt

, (1.17)

where ∆ is a real symmetric matrix and Π is a spin independent Hermitian matrix.The conservation of the total spin allows to determine the diagonal elements ∆pp =n2p and Πpp = np [67], whereas known analytical necessary N -representability condi-

tions provide bounds for the off-diagonal terms [68]. In detail, the N -representabilityD and Q conditions of the 2-RDM impose the following inequalities on the off-diagonal elements of ∆:

∆qp ≤ nqnp, ∆qp ≤ hqhp, q 6= p (1.18)

and similarly, condition G imposes the next inequality on the Π-matrix:

Π2qp ≤ (nqhp + ∆qp) (hqnp + ∆qp) , (1.19)

where hp = (1− np) is the hole in the spatial orbital p. It is worthwhile pointingout that according to Eq. (1.19) the signs of Πqp are indeterminate, so the signconvention has to be determinated arbitrarily. Furthermore, due to the contractionrules of the two-matrix (1.8), ∆ must fulfill

∑′q∆qp = nphp, (1.20)

where the prime indicates hereafter that the q = p term is omitted from the sum-mation. So far, the Bottom − Up method proposed by Piris leads to the next energyexpression for singlet state systems

E =∑p

np (2Hpp + Jpp) +∑pq

′ΠqpLpq +∑pq

′ (nqnp −∆qp) (2Jpq −Kpq) , (1.21)

11


where Hpp is the matrix element of the kinetic energy and nuclear attraction terms,whereas Jpp =< pp|pp > is the Coulomb interaction between two electrons withopposite spins at the same spatial orbital p. Jpq = 〈pq|pq〉 and Kpq = 〈pq|qp〉 arethe usual direct and exchange integrals, respectively. Lpq = 〈pp|qq〉 is the exchangeand time-inversion integral [57], which reduces to Kpq for real orbitals. Note thatthe energy expression (1.21) belongs to the familiy of JKL-only family of NOFs.Different approximations to off-diagonal terms of matrices ∆ and Π lead to thefamiliy of PNOFi, i=1, 6 [60, 61].In this work, we employ the PNOF6 [61], which has proved a better treatment ofboth dynamic and non-dynamic correlations than its predecessors [71, 65, 40, 6, 63].We focus on the extended version of PNOF [66], which provides a more flexibledescription of the electron pairs in the NOF framework. PNOF6 is an orbital-pairingmethod, i.e. each orbital g below the Fermi level (F = N/2) couples with Nc orbitalsabove it (p > F ), so the space is divided into disjoint orbital subspaces such Ωg ≡g, p1, p2, · · · , pNc . Taking into account the spin, each subspace contains an electronpair. Henceforth, we will denote PNOF6(Nc) the method we use, emphasizing thenumber Nc of usually weakly-occupied orbitals employed in the description of theelectron pairs. Due to (1.20) the occupancies corresponding to each subspace Ωg

must fulfill the relation

∑p∈Ωg

np = 1 , g = 1, F . (1.22)

It is noteworthy that the reconstruction of the 2-RDM, and therefore the functional(1.21), are independent of the orbital-pairing sum rules (1.22). These additionalconstraints are imposed to ensure that no fractional electron numbers appear whennon-dynamic electron correlation effects become important [62, 45, 73]. Additionally,this allows the constraint-free minimization of the PNOF6(Nc) energy with respectto the occupation numbers, which yields substantial savings of computational time.So far, we have presented the keys to construct a Piris Natural Orbital Functionalbased on the orbital-pairing approach, so we have just to specify the form of themethod employed in this work, PNOF6(Nc). The PNOF6(Nc) energy for a singletstate of an N -electron molecule can be cast as

E =F∑g=1

Eg +F∑f 6=g

∑p∈Ωf

∑q∈Ωg

Eintpq . (1.23)

The first term of the energy (1.23) draws the system as independent F = N/2electron pairs described by the following NOF of two-electron systems,

Eg =∑p∈Ωg

np (2Hpp + Jpp) +∑

p,q∈Ωg ,p 6=qEintpq . (1.24)

12

1.1 The NOF Theory

It is worth noting that the interaction energy, the last term of equations (1.23) and(1.24), is equal for electrons belonging to the same subspace Ωg or two differentsubspaces (Ωg 6= Ωf ), as correspond respectively the one in Eq. (1.24) and theother in Eq. (1.23). Therefore, the intrapair and interpair correlations are equallybalanced in PNOF6(Nc). Because of this feature PNOF6 is able to describe properlysystems with several delocalized electrons, such as Benzene or H4 [61, 71]. Theinteraction energy between electrons Eint

pq is given by

Eintpq = (nqnp −∆qp) (2Jpq −Kpq) + ΠqpLpq. (1.25)

Recall that PNOF6 goes beyond the independent pair approximation, due to thecontribution that comes from those ∆qp related to orbitals belonging to differentpairs. As explained above, having appropiate forms for the off-diagonal elementsof matrices ∆ and Π is crucial to describe correctly the electronic structure of anyatomic or molecular system, somehow, the quality of a PNOF depends on thisdecision. In the case of PNOF6(Nc), the off-diagonal terms of ∆ and Π matrices are

∆qp Πqp Orbitals

e−2Shqhp −e−S (hqhp)1/2 q ≤ F, p ≤ F

γqγpSγ

−Πγqp

q ≤ F, p > Fq > F, p ≤ F

e−2Snqnp e−S (nqnp)1/2 q > F, p > F

, (1.26)

where

γp = nphp + α2p − αpSα

αp =e−Shp , p ≤ F

e−Snp , p > F

Πγqp =

(nqhp + γqγp

Sγ

)1/2 (hqnp + γqγp

Sγ

)1/2

S =F+FNc∑q=F+1

nq, Sα =F+FNc∑q=F+1

αq, Sγ =F+FNc∑q=F+1

γq

. (1.27)

Remark that S -type factors can be equally defined as S = ∑F1 hq and so on. These

factors play an important role in PNOF6. When the holes below the Fermi level arenot very significant, S takes small values, so e−2S ≈ e−S ≈ 1 and the form of PNOF6simplifies to previous functionals of this family that use the simplest formulation of

13


αp. Nevertheless, for large values of S the exponential factor decreases rapidly,thereby in the limit αp tends to zero and γp = nphp, so that N -representability Dand Q conditions are fulfilled quite simply, whereas the Π matrix is constructedsuch that the G condition is fulfilled too. Recall that PNOF6 has been proved tobe able to compete with other well-established high-level electronic structure meth-ods. Thus, it has been tested for dissociation curves of diatomic and non-diatomicmolecules [71], single-point energies, equilibrium distances, harmonic frequencies,and anharmonicity constants [61, 64, 42, 65, 40, 6, 63].Finally, let us give some notes about how afford the optimization of PNOF in orderto solve the many electron problem. The solution in NOFT is established optimiz-ing the energy functional with respect to the ONs and the NOs separately. PNOF6allows constrain-free minimization with respect to the ONs just by employing thecosine function to calculate them, i.e ∀p np = cos2 (γp), then parameters γp canvary without any constraint. Conversely, orthonormality of the NOs (⟨φi|φj⟩ = δpq)has to be forced when minimizing the energy with respect to the real orbitals, andhere is the bottleneck of the optimization process. At present, the procedure for theminimization of the energy (1.23) with respect to both the occupation numbers andthe natural orbitals is carried out by the iterative diagonalization method developedby Piris and Ugalde [70], which is the more efficient procedure for the currentlyNOFs in use [5]. Essentially, this method consists in yielding the natural orbitalsby iterative diagonalization of a Hermitian matrix F , whose off-diagonal elementsare determined explicitly from the hermiticity of the matrix of the Lagrange multi-pliers [59], whereas its diagonal elements are determined from an aufbau principle.Since there is no a formal expression for the diagonal elements, F is not strictly ageneralized Fockian.

1.2 Molecular Electrostatic Moments

The complete determination of the whole charge density and various response func-tions (and the associated deformation densities) is computationally heavy and con-ceptually not very rewarding. As any distribution function, the essential features ofthe charge distribution can be characterized by its moments. Consider an arbitrarycharge distribution ρ (r′), which is nonvanishing only inside a sphere of radius Raround some origin. The electrostatic potential outside the sphere can be writtenas a series expansion in spherical harmonics [33]:

V (r) =∞∑l=0

l∑m=−l

12l + 1qlm

Ylm (θ, φ)rl+1 . (1.28)

This equation is called a multipole expansion; so that the l = 0 term is called themonopole term, l = 1 are the dipole terms, etc. In practice, the use of spherical

14


harmonics in the expansion is useful when the charge distribution is spherical ornearly spherical. If we evaluate the potential outside the charge distribution, then(1.28) simplifies to

V (r) =∑l,m

12l + 1

[ˆY ∗lm (θ′, φ′) r′lρ (r′) d3r′

]Ylm (θ, φ)rl+1 , (1.29)

where the coefficients qlm =[´Y ∗lm (θ′, φ′) r′lρ (r′) d3r′

]are called multipole moments.

In molecular physics, the multipolar expansion of the electrostatic field is useful todetermine long range intermolecular interactions, because in this case a few termsof the series expansion are sufficient to deal with the problem. In practice, it is notalways convenient to work with spherical harmonics, hence in the context of classicalelectrodynamics it is more common to use the Coulombic expression to derive themultipole moments. Let us consider again an arbitrary charge distribution ρ (r′),limiting r′ to the space where the charge distribution extends and using the fact thatr r′, i.e the potential is evaluated at large distances from the charge distribution,we can carry out a Taylor series expansion of 1/|r−r′| in the integral form of theelectrostatic potential:

V (r) = 1r

ˆρ (r′)

1− r · r′

r+ 1

2r2

(3 (r · r′)2 − r′2

)+O

(r′

r

)3 d3r′. (1.30)

Molecular multipole moments arise from equation (1.30) selecting consecutivelyterms from the expansion

(1− r·r′

r+ 1

2r2

(3 (r · r′)2 − r′2

)+O

(r′

r

)3). Thus, if we

want to approximate this potential up to quadrupole order, then we can just truncateevery term which scales worse than r−3, and so on. According to both expressions(1.29) and (1.30), the multipole expansion is expressed as a sum of terms withprogressively finer angular features, i.e, high-order terms vary quickly with angles,whereas the monopole (0th order) is scalar and the dipole moment varies once frompositive to negative around the whole sphere. Thereby, electric moments measurethe departure from spherical symmetry of a charge distribution.In the framework of quantum mechanics, an alternative and more appropiate formfor defining molecular electric moments consists in introducing what is called theelectrostatic multipole operator

Qml ≡

N∑i=1

eZiRml (ri) , where Rm

l (r) ≡√

12l + 1r

lY ml (θ, φ) . (1.31)

Rml (ri) is a regular solid harmonic function in Racah’s normalization, which is a

solution of the Laplace equation in spherical polar coordinates that vanishes at the

15


origin. Then, if the molecule is characterized by a normalized wave function ψ, themultipole moment of order l corresponding to that molecule is simply given by theexpectation value

Mml ≡ ⟨ψ|Qm

l |ψ⟩. (1.32)

From equation (1.32) is straightforward to conclude that there is a direct relationbetween molecular electric moments and the point group symmetry of the molecule.In the end, note that depending on the symmetry of the wave function and theparity of the operators Qm

l the corresponding multipole moment of order l will bezero or not.

So far, we have discussed the different mathematical definitions of electrostatic mul-tipole moments, let us now focus on the role that play these properties for atomicand molecular physics, also known as modern quantum chemistry. The calculationof molecular electric moments is of both practical and fundamental interest. Atfirst, we have already seen that molecular electric moments give information aboutthe wave function, therefore, the theoretical calculation of these properties is use-ful to test the method employed beyond the energy, concerning the quality of thecomputed wave function or reduced density matrices, and the corresponding chargedistribution (chemical bonds...). Thus, comparing the calculated electric momentswith experimental data or values obtained using well-established high-level elec-tronic structure methods such as CCSD(T), we can test the predictive quality ofany calculation method. Regarding the practical interest, we can distinguish be-tween two main applications. On the one hand, at large distances intermolecularforces are completely determined by permanent multipole moments, so dispersioninteractions, responsible of many phenomena in chemistry and hard to parametrizein computer simulations, can be determined using this method. Similarly, the in-teraction between a molecule and an external field can be characterized simply bysome terms of the electrostatic expansion. So multipole moments help calculatingthese quantities and thereby understanding properties of imperfect gases, liquids,and solids. On the other hand, as we will explain in more detail later, although thereexist several experimental techniques in order to obtain molecular dipole moments[22, 48], it is still a challenge to obtain accurately and independently of experimen-tal conditions molecular quadrupole, octupole and higher order moments [3, 14, 8].That is why theoretical calculations are helpful in the determination and predictionof these properties.

It is worth pointing out that there are different forms for defining electric multi-pole moments [23] (except in the case of monopole and dipole moments) associatedto atomic and molecular systems. The convention of Buckingham [3, 4] is usedthroughout this work, then, only the first non-zero term of the series expansion isindependent of the origin of coordinates, which is commonly the center of mass.

16


Buckingham defines the multipole moments as traceless, so that they form an irre-ducible representation of the rotation group. In essence, doing a multipole expansionis about estimating the potential of the system on a sphere of large radius r, thatis why you want the functions arising from the expansion to embody representa-tions of the rotation group. As it turns out, all the potentials in a given irreduciblerepresentation must have the same dependence on r, then, due to having the sameangular dependence, you can rotate any function within the representation subspaceinto any other, in the same way you can rotate a z-pointing dipole into an x-pointingone. Thus, irreducible representations are forced by the spherical symmetry to havethe same radial dependence throughout, whereas the presence of invariant subspaceswould break this property, which is why we use notation to explicitly take those out.Formally, the general definition of traceless (Buckingham) multipole moments isgiven by

M(n)α,β...ν = (−1)n

n!

ˆdr′ρ (r’) r′2n+1 ∂n

∂r′α∂...r′ν

( 1r′

). (1.33)

This formula is equivalent to previous multipole expressions as (1.32) in the sensethat the electrostatic potential can be directly calculated from them, in the case of(1.33) using the formula

V (r) =∑n

(−1)n

(2n− 1)!!M(n)α,β...ν∇α∇β...∇(n)

ν

(1r

). (1.34)

So far, we have discussed the origin of electrostatic multipole moments, as well as thecorresponding general expressions and their practical and fundamental interest. Inthe next sections, the most important electric moments such as dipole, quadrupoleand octupole moments, are studied in detail.

1.2.1 Dipole Moment

The dipole moment or 2nd order multipole moment is the first non-scalar term ofthe electrostatic multipole expansion. It is a three component vector, which in theconvention of Buckingham [3, 4] is expressed by the formula

µα = −12

ˆρ(r)rαdV +

NUC∑i=1

ZiRiα, (1.35)

where the Greek index denotes the Cartesian directions x , y and z . Note that thenuclear contribution is taken into account separately from the electronic contribu-tion. Thus, as usually done in quantum chemistry, the atomic nuclei are considered

17


as points in the space, what is called the point-charge approximation, whereas thenegative charge is spread over all the system yielding what is known as charge distri-bution, expressed by the electronic density ρ(r). Qualitatively, the dipole momentof an uncharged body can be thought of as being formed by separating positive andnegative charges, the magnitude of the dipole being the product of the charge andthe separation.The principal experimental methods of dipole moment measurement are MicrowaveSpectroscopy (MW), Molecular Beam Electric Resonance (MB), Dielectric (DT, DR,NR), Temperature-Variation Procedure (DT), Indirect (Optical) Procedure (DR),and Nonresonant Microwave Absorption or Dispersion (NR). Dipole moments de-termined by spectroscopic methods always refer to a particular vibrational and elec-tronic state and to a single isotopic species, while in contrast moments determinedfrom measurements of bulk dielectric properties represent an average over the equi-librium population of vibrational and electronic states, and over the natural abun-dance of the various isotopic species. Compared to the spectroscopic techniques, thedielectric methods have certain limitations and disadvantages. For more details see[52].Programming and computation of the molecular dipole moment has been done in theDoNOF program package developed by Piris. Details can be found in APPENDIXI.

1.2.2 Quadrupole Moment

The molecular electric quadrupole moment is a property of special interest for non-polar molecules, due to being the first non-vanishing term of the electrostatic multi-pole expansion in these cases. Thereby it is the main contribution to the electrostaticinteractions either with other molecules or with non-uniform external electric fields.Regarding the physical interpretation, the quadrupole moment of a system with zerodipole moment can be thought of as arising from a separation of equal and oppositedipoles, the magnitude of the quadrupole being proportional to the product of thedipole moment and the separation. Structurally, the absence of a quadrupole mo-ment must mean that CH4 is tetrahedral. Anyway, these interpretations are limitedto a few cases, and as happen in general with high-order multipole moments, thephysical interpretations are complex, and limited to specific cases.Let us now derive the traceless quadrupole moment of Buckingham. The potentialof a quadrupole is

V (2) (r) = 12Qαβ∇α∇β

(1r

), (1.36)

where Qαβ =´rαrβρ (r) dr, and the superindex (2) denotes the 2nd term of the

18


general electrostatic potential. Note that Laplace equation is satisfied

∇2(1r

)=∑α

∇α∇α

(1r

)= 0. (1.37)

We can add to the quadrupole potential an arbitrary quantity without changing itsvalue according to (1.37),

V (2) (r) = 12 (Qαβ + λδαβ)∇α∇β

(1r

). (1.38)

Then, setting the value of λ equal to the trace of Qαβ, i.e λ = −13Qαα = −1

3∑αQαα,

the new expression of the quadrupole potential is

V (2) (r) = 12 ·

13

ˆρ(r′)(3r′αr′β − δαβr′2)∇α∇β

(1r

)dV (1.39)

Thus, taking into account the nuclear contribution, not explicitly written in previousexpressions of this section, the traceless (Θαα = 0) quadrupole moment is defined inthe convention of Buckingham [3, 4] as

Θαβ = −12

ˆρ(r)(3rαrβ − δαβr2)dV+ 1

2

NUC∑i=1

Zi(3RiαRiβ − δαβR2i ) . (1.40)

Greek indices denote the Cartesian directions x , y and z . Equation (1.40) defines asymmetric tensor in all sufixes. This convention is specially useful when computingthe electrostatic energy arising from the interaction between quadrupole momentand the field gradient, which according to Laplace’s equation fulfills

F′

αα = F′

xx + F′

yy + F′

zz = 0, where F ′

αβ = −(

∂2φ

∂rα∂rβ

). (1.41)

The most popular experimental techniques to obtain molecular quadrupole momentsare Collision-Induced Spectroscopy and Electric Field Gradient Induced Birefrin-gence. However, the former depends on the intermolecular potential model used toanalyze the data, and the latter is temperature dependent and cannot be appliedto all systems. Another experimental technique is the molecular beam Zeemanmethod, though it presents some drawbacks too, because it can only be used if themagnetizability χ and the rotational g factor are known. Overall, experimentalvalues of quadrupole moments are not unique and reliable, and still today presentsdependence on other equally elusive properties. For more details see [34, 14, 50].Programming and computation of the molecular quadrupole moment has been donein the DoNOF program package developed by Piris. Details can be found in AP-PENDIX I.

19


1.2.3 Octupole Moment

As one goes to higher order terms in the multipole expansion of the electrostaticpotential the practical and fundamental interest decreases. From the mathematicalpoint of view, the Taylor expansion (1.30) in powers of 1/|r−r′| goes to zero evalu-ating the potential at large distances from the charge distribution, i.e 1/|r−r′|α 0for large α values due to the assumption r r′. That is why the series are com-monly truncated and only the first few terms of the expansion are kept, usuallyup to quadrupole or octupole in molecular science. Thus, the octupole moment isparticularly interesting when it is the first non-zero term of the electric multipoleexpansion, for example, in the case of methane or similar tetrahedral molecules. Infact, the octupole-octupole interaction is the main long-range orientation dependentinteraction in methane, and what is more, dispersion energy terms are fundamentalin the binding energy of materials containing tetrahedral molecules.The octupole moment is defined by Buckingham [3, 4] as

Ωαβγ = −12

ˆρ(r)

[5rαrβrγ−r2 (rαδβγ + rβδαγ + rγδαβ)

]dV

−12

NUC∑i=1

Zi[5RiαRiβRiγ −R2

i (Riαδβγ +Riβδαγ +Riγδαβ)] . (1.42)

Greek indices denote the Cartesian directions x , y and z . Equation (1.42) defines asymmetric tensor in all suffixes. Also, (1.42) leads to Ωxxz + Ωyyz + Ωzzz = 0 foroctupole tensor, and respective permutations between the subscripts x , y and z . Insummary, Ωααβ = 0.Regarding the experimental determination of molecular octupole moments, one isforced to use indirect methods based on accessible measurements of effects reveal-ing perceptibly interactions of the octupolar type, for instance, dipole-octupole,quadrupole-octupole, or octupole-octupole interactions. In the case of tetrahedralmolecules, that are the most interesting regarding the investigation of this property,it is usual to investigate the second virial coefficients of appropiately chosen gasmixtures, that are directly related to the octupole moment.Programming and computation of the molecular octupole moment has been donein the DoNOF program package developed by Piris. Details can be found in AP-PENDIX I.

20

2 Methodology

In this section we describe the strategy used in order to carry out the PNOF6(Nc)calculations, i.e the details of the calculations such as the geometry or the basis setemployed. Also, the methodology used to compare our results with experimentalmarks and theoretical benchmark calculations is commented.

First, regarding the basis set used to carry out the calculations, it is known to bean important factor in the calculation of molecular electric properties. As recentlyshown by Wendland and Hofinger [29], the proper description of the outer-valenceregion is crucial in order to carry out a quantitative analysis, so the chosen basis setmust be sufficiently large in order to contain augmented functions that are necessaryto obtain accurate values of molecular electric moments. We used the Gaussianbasis set of Sadlej [75, 76], which is a correlation-consistent valence triple-ζ basisset augmented with additional basis functions selected specifically for the correlatedcalculation of electric properties. In detail, this set consists of (6s4p) Gaussian-typefunctions (GTF) contracted to [3s2p] on H, (10s6p4d) GTF contracted to [5s3p2d]for C to F, and (13s10p4d) GTF contracted to [7s5p2d] for Si to Cl. Thus, it containssufficient diffuse and polarization functions in order to give an accurate descriptionof the outer-valence region. The Sadlej basis set has effectively the same accuracyas the aug-cc-pVTZ basis set [27].

Since the main aim of this work is to test the predictive quality of PNOF6(Nc) com-puting molecular electric moments, we focus on comparing PNOF6(Nc) results withexperimental data. Therefore, all calculations were performed at experimental equi-librium geometries taken from [54, 34, 31, 28]. Experimental dipole and quadrupolemoments were taken from Refs. [54, 34, 2, 13, 25, 24, 37, 16, 44, 74] unless otherwisestated. We made no correction for vibrational effects because they are negligible forthe studied set of molecules; nevertheless, as pointed out by Russel and Spackman[74], these corrections may be important for molecules containing small multipolemoments. Therefore, it is useful to show molecular multipole moments obtained withanother high-level electronic structure method in order to carry out a more completecomparison. We have included the calculated Hartree-Fock (HF) and Coupled Clus-ter Singles and Doubles (CCSD) values obtained by us using the GAMESS program[77, 21]. Recall that the CCSD results for one-electron properties differ from full-CIresults in a 2% [1], actually, the excellent performance of the CCSD relies on thefact that the chosen molecules, at least those for which CCSD is used for compar-ison, are all dominated by a single configuration. Hence, the CCSD values can beconsidered as benchmark calculations within the numerical accuracy of this work.

21

Chapter 2 Methodology

Nonetheless, in systems with a high multiconfigurational character, i.e where severalSlater determinants or configurations are needed to describe properly the wave func-tion of the molecular system, the performance of CCSD is quite bad due to beinga single-reference method. That is why for these cases the Multi-Reference Singlesand Doubles - Configuration Interaction (MRSD-CI) method is used as benchmarktheoretical method [54]. Naturally, all CCSD and MRSD-CI calculations are carriedout at experimental equilibrium geometries using the Sadlej-pVTZ basis set. Theperformance of theoretically obtained results is established by carrying out a statis-tical analysis of the mean absolute errors (MAE) with respect to the experimentaldata. Formally, the MAE is a quantity used to measure how close predictions areto the eventual outcomes, and is given by

MAE = 1n

n∑i=1|fi − yi|, (2.1)

where fi is the prediction, in our case the theoretically obtained values, and yi thetrue value, that in this work consists of experimental values.We choose a wide selection of spin compensated molecules composed of main groupelements. We apply PNOF6, in its extended version, to the determination of molecu-lar dipole and quadrupole moments of H2, HF, BH, HCl, H2O, H2CO, C2H2, C2H4,C2H6, C6H6, CH3CCH, CH3F, HCCF, ClF, CO, CO2, O3, N2, NH3, and PH3.Moreover, the octupole moment of CH4, a molecule without dipole and quadrupolemoments is also studied. Is worth noting that the matrix element of the kineticenergy and nuclear attraction terms, as well as the electron repulsion integrals areinputs to our computational code. Thus, in the current implementation, we haveused the GAMESS program [77, 21] for this task. For the sake of simplicity, atomicunits (a.u.) are used throughout.

22

3 Results and Discussion

In the following sections, we show the PNOF6(Nc) results obtained for molecularelectric moments with respect to the center of mass. The number Nc of usuallyweakly-occupied orbitals employed in the description of the electron pairs variesdepending on each molecular system. For comparison, we have included the availableexperimental data, the calculated Hartree-Fock (HF) and CCSD values using theGAMESS program [77, 21], and the MRSD-CI values [54]. We discuss the outcomesobtained for the dipole, quadrupole and octupole moments, in separate sections.

3.1 Dipole moment

In this work, the dipoles are aligned along the principal symmetry axis of the studiedmolecules, set on z direction. Tab. 3.1 shows the independent component µz ofthe dipole moments obtained at the HF, PNOF6(Nc), and CCSD levels of theory.Accordingly, the MAE values obtained for each molecular system by using differentmethods are plotted in Fig. 3.1.Overall, the inclusion of electron correlation effects through, both PNOF6(Nc) andCCSD, improves significantly the performance of the HF method. Thus, lookingat Fig. 3.1, the yellow bars, which correspond to the HF MAEs, are much morehigher than those corresponding to both CCSD and PNOF6(Nc) results, that arerepresented by the red and blue bars respectively. Quantitatively, PNOF6(Nc) andCCSD afford MAEs with respect to experimental data of 0.0289 a.u. and 0.0177a.u., respectively. Hence, it is worth noting the agreement between PNOF6(Nc)and CCSD results, as well as with the experimental data, so the obtained dipolemoments are numerically accurate.Note that the aug-cc-pVTZ basis set of Dunning [11] was used for the BH moleculesince there is no Sadlej-pVTZ basis set available for Boron. In this case, thePNOF6(1) result is very close to the Full-CI/aug-cc-pVTZ value obtained by Halkieret al. [1], 0.5433 a.u., showing a result as good as the CCSD one. Similarly, theagreement between PNOF6(3) and Full-CI/aug-cc-pVTZ is excellent for HF, in fact,the value reported in Ref. [1] of 0.7025 a.u. differs from our result only in a 0.2%.Comparing to the experimental value, the error is below 0.01 a.u. as shown inTab. 3.1.The electronic structure and bonding situation of carbon monoxide is of specialinterest for modern electronic structure methods. The dipole moment of CO, ex-

23

Chapter 3 Results and Discussion

Table 3.1: µz component of molecular dipole moments in atomic units (ea0) com-puted with the Sadlej-pVTZ basis set at the experimental equilibrium geometries[31]. Nc is the number of weakly-occupied orbitals employed in PNOF6(Nc) foreach molecule.

Molecule HF PNOF6 (Nc) CCSD EXP.HF 0.7565 0.7009 3 0.6994 0.7089 [54]BH∗ 0.6854 0.5574 1 0.5551 0.4997 [54]H2O 0.7808 0.7394 1 0.7225 0.7268 [54]H2CO 1.1134 0.9480 1 0.9084 0.9175 [54]HCl 0.4746 0.4553 1 0.4416 0.4301 [54]HCCF 0.3535 0.3434 5 0.2733 0.2872 [14]NH3 0.6372 0.5968 1 0.5943 0.5789 [34]PH3 0.2780 0.2635 1 0.2340 0.2258 [29]O3 0.3033 0.1182 1 0.2276 0.2099 [44]ClF 0.4453 0.3226 6 0.3451 0.3462 [13]CH3F 0.7706 0.7278 5 0.6919 0.7312 [74]CH3CCH 0.3203 0.2996 5 0.2866 0.3070 [2]CO −0.0987 0.0444 5 0.0725 0.0481 [54]MAE 0.0843 0.0289 0.0177

∗Calculations performed with the aug-cc-pVTZ basis set.

tensively studied in Refs. [15, 35, 43], is very small (0.0481 a.u.) and ends at thecarbon atom, although carbon is less electronegative than oxygen. The result shownin Tab. 3.1 is representative, while HF gives the wrong direction for the CO dipolemoment, PNOF6(5) corrects the sign, giving a result that is in excellent agreementwith the experimental value. Remarkably, the result obtained at CCSD level is34% away from the experimental value, so that it is necessary to include third ordertriplet excitations in the cluster theory in order to obtain a reasonable value, such asthe one reported by Maroulis [43] at the CCSD(T) level, 0.0492 a.u.. Accordingly,the relevant electron correlation for CO is well accounted by the PNOF6(5) method.

Regarding the values obtained for H2O, H2CO, HCl, NH3, and ClF, PNOF6(Nc)competes with Coupled Cluster, providing values that differ from experimental datain less than a 5%. In the case of HCCF and PH3, PNOF6(Nc) seems to lack rel-evant dynamic electron correlation and thereby the obtained dipole moments arenot as accurate as the CCSD ones. Conversely, PNOF6(5) is in excellent agreementwith experimental data in the case of the methyls CH3F and CH3CCH, often at-tached to large organic molecules, giving dipole moments with errors of 0.5% and2% respectively, with respect to experimental values.

A special case is ozone, which is a molecule with strong multiconfigurational charac-ter. Indeed, paying attention to the occupations obtained in the PNOF6 calculation

24

3.2 Quadrupole moment

Figure 3.1: Obtained MAE values for dipole moments.

0

0.05

0.1

0.15

0.2

Dipole moments errors (a.u.) with respect to EXP

HF BH H2O H

2CO HCl HCCF NH

3PH

3O3 ClF CH

3F CH

3CCH CO

HFPNOF6CCSD

of ozone, there is one spatial orbital below the Fermi level with occupation 1.63568,and correspondingly, there is one spatial orbital above Fermi level with occupation0.36432. Thus, two Slater determinants are at least necessary to describe properlythe wave function of this molecule. In other words, O3 is a typical two-configurationsystem. Regarding the calculations, the PNOF6(1) dipole moves into the right di-rection from the HF value, but overestimates the effects of the electron correlation.Taking into account the good CCSD result for O3, which is not valid for higherelectric moments, it seems that the dynamic electron correlation compensates forthe lack of non-dynamical in this method, and could improve our numerical valueof the dipole.


Tab. 3.2 and Tab. 3.3 list the molecular quadrupole moments obtained at the HF,CCSD, MRSD-CI and PNOF6(Nc) levels of theory, along with the experimentalvalues taken from Refs. [54, 34, 13, 14, 44, 74, 29, 25, 16, 53, 10]. Accordingly,the MAE values obtained for each molecular system by using different methods areplotted in Fig. 3.2 and Fig. 3.3. Inspection of these Tables shows that PNOF6(Nc)quadrupole moments agree satisfactorily with the experimental data, whereas thediscrepancies are consistent with those observed using the CCSD and MRSD-CImethods in most cases. Regarding Fig. 3.2 and Fig. 3.3, the trend is similar to theone observed in Fig. 3.1, while yellow bars stand out due to the poor performance of

25


HF method, inclusion of correlation effects lead to smaller MAE values, so that redand blue bars are notably lower. Nevertheless, relative differences are slightly smallerthan in the case of dipole moments because of the complexity of the quadrupolemoment, even more when several components of the quadrupole tensor are studied,as shown in Fig. 3.3.

Table 3.2: Θzz component of the quadrupole moments, in atomic units, computedwith the Sadlej-pVTZ basis set at the experimental equilibrium geometries [31]for molecules with linear, C3v, D6h, and D3d symmetry. Nc is the number ofweakly-occupied orbitals employed in PNOF6(Nc) for each molecule.

Molecule HF PNOF6 (Nc) CCSD EXP.H2 0.4381 0.3937 17 0.3935 0.39± 0.01 [53]HF 1.7422 1.6978 3 1.7156 1.75± 0.02 [34]BH∗ 2.6772 2.3303 1 2.3388 2.3293† [1]HCl 2.8572 2.7812 1 2.7233 2.78± 0.09 [54]HCCF 3.3530 3.2581 5 2.9335 2.94± 0.10 [14]CO 1.5366 1.4237 1 1.4889 1.44± 0.30 [34]N2 0.9397 1.1195 1 1.1712 1.09± 0.07 [34]NH3 2.1258 2.1048 3 2.1661 2.45± 0.30 [54]PH3 1.7217 1.6488 1 1.5695 1.56± 0.70 [29]ClF 0.9413 1.1122 6 1.0514 1.14± 0.05 [13]CH3F 0.3482 0.3294 5 0.3002 0.30± 0.02 [74]C2H2 5.3655 5.1073 1 4.6850 4.71± 0.14 [10]C2H6 0.6329 0.6233 5 0.6234 0.59± 0.07 [16]C6H6 6.6418 6.3542 1 5.6653 6.30± 0.27 [25]CH3CCH 4.2913 4.0892 5 3.6939 3.58± 0.01 [31]CO2 3.8087 3.5625 1 3.1966 3.19± 0.13 [34]MAE 0.2646 0.1418 0.0902

∗Calculations performed with the aug-cc-pVTZ basis set.† Full CI calculation reported by Halkier et al. [1]

In the case of linear molecules (H2, HF, BH, HCl, HCCF, ClF, CO, C2H2, CO2and N2), NH3 and PH3, belonging to the C3v point symmetry group, the D6h C6H6molecule, and the trigonal planar C2H6, which hasD3d symmetry, the relation Θxx =Θyy = −1

2Θzz holds for quadrupole moment tensor, so Θzz alone is sufficient todetermine completely the quadrupole moment. Setting the main axis of symmetry inthe z direction of the coordinate system, the results for these molecules are reportedin Tab. 3.2. From the latter, one can observe that PNOF6(Nc) yields a MAE of0.14 a.u., hence considering the added complexity of the quadrupole moment, theperformance of PNOF6(Nc) is within a reasonable accuracy.Taking into account the experimental uncertainty, PNOF6(Nc) results agree withthe experimental data for H2, HCl, CO, N2, PH3, ClF, CH3F, C2H6, and C6H6.

26


Figure 3.2: Obtained MAE values for symmetric quadrupole moments.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Quadrupole moments errors (a.u.) with respect to EXP

H2 BH HF HCl HCCF CO N

2NH

3PH

3 ClF CH3F C

2H2C2H6C6H6CH

3CCH CO

2

HFPNOF6CCSD

The value obtained for H2 reproduces the experimental one with high precision.It is also worth noting the excellent agreement with the experiment obtained forthe quadrupole moment of Benzene, which is of great interest for many fields ofchemistry and biology [25, 36]. Indeed, the quadrupole moment of Benzene playsan important role in determining the crystal structures and molecular recognition inbiological systems because it is the key to the intermolecular interactions betweenπ-systems.

For HCCF, NH3, C2H2, and CO2, the quadrupole moments fall out of the exper-imental error intervals, however, in the case of HCCF, C2H2, and CO2 the meanrelative percentage error is below 10%, whereas the results obtained for NH3 is only0.05 a.u. away from the higher limit of the experimental uncertainty. For CH3CCHthe PNOF6(5) result deviates from the experimental value in a 12%, ergo more dy-namic correlation is clearly necessary to improve this result, an effect not observedfor the dipole moment of this molecule.

For the Hydrogen fluoride, the HF result is the closest to the experimental value,however, the PNOF6(3) result is in outstanding agreement with the full-CI/aug-cc-pVTZ value of 1.6958 a.u. [1]. For the Boron monohydride, the experimentalquadrupole moment is not available, so we use the full-CI/aug-cc-pVTZ calculationreported by Halkier et al. [1], 2.3293 a.u., in order to carry out the comparison.The agreement between PNOF6(1) and full-CI is as good as in the case of Hydrogenfluoride, according to the relative percentage error obtained below 0.05%.

Tab. 3.3 shows the Θzz and Θxx components obtained for H2O, H2CO, C2H4, and

27


Table 3.3: Θzz and Θxx components of molecular quadrupole moments, in atomicunits, computed using the Sadlej-pVTZ basis set at the experimental equilib-rium geometries [31]. Nc is the number of weakly-occupied orbitals employed inPNOF6(Nc) for each molecule.

Molecule HF PNOF6 (Nc) MRSD-CI EXP.H2O (xx) 1.7966 1.7808 9 1.8050 1.86± 0.02 [54]H2O (zz) 0.0981 0.0869 9 0.0950 0.10± 0.02 [54]H2CO (xx) 0.1019 0.0310 1 0.1100 0.04± 0.12 [37]H2CO (zz) 0.0921 0.1432 1 0.2230 0.20± 0.15 [37]C2H4 (xx) 2.7819 2.5766 3 2.3700 2.45± 0.12 [54]C2H4 (zz) 1.4942 1.3204 3 1.1700 1.49± 0.11 [54]O3 (xx) 1.1175 1.2522 1 1.2830 1.03± 0.12 [44]O3 (zz) −0.2387 0.4204 1 0.1680 0.52± 0.08 [44]MAE 0.1772 0.0970 0.1448

O3. In this work, we use the traceless quadrupole moment, hence two componentsare sufficient to determine completely this magnitude. On the other hand, MRSD-CI values are significantly better than CCSD calculations when many componentsof the quadrupole tensor are studied [54], thereby MRSD-CI is used as benchmarktheoretical method in Tab. 3.3.

According to the results reported in Tab. 3.3, PNOF6(Nc) performs better thanthe MRSD-CI method for this selected set of molecules. For H2O and H2CO, thePNOF6(Nc) values fall into the experimental error interval, which is specially broadfor H2CO. In the case of the C2H4 molecule, the longitudinal component Θzz ob-tained with PNOF6(3) increases with respect to the HF value, whereas the Θxx

component decreases. Finally, we have the results obtained for O3, which is a strin-gent test for quadrupole calculations due to its two-configurational character [44, 78].One can observe that Ozone is well described by PNOF6(1) comparing to the resultsobtained by using HF and MRSD-CI methods. In fact, regarding the most relevantcase, HF gives the wrong sign for the zz−component of ozone quadrupole, andMRSD-CI poorly corrects it, giving a value with a relative percentage error of 68%.It is worth mentioning that CCSD gives a value even worse than MRSD-CI for thiscomponent, 0.1658 a.u.. Conversely, the value obtained by using PNOF6(5) is inapproximate agreement with the experiment if the experimental error is taking intoaccount. Thereby, the O3 molecule, which has long been considered to be a demand-ing test case for quantum chemical methods, is well described by PNOF6(Nc) dueto treating in a balanced way both dynamic and non-dynamic electron correlations.

28

3.3 Octupole moment

Figure 3.3: Obtained MAE values for non-symmetric quadrupole moments.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Quadrupole moments errors (a.u.) with respect to EXP

O3(zz)O3(xx)C2H4(zz)C2H4(xx)H2CO(zz)H2CO(xx)H2O(zz)H2O(xx)

HFPNOF6MRSD

3.3 Octupole moment

The octupole moment is particularly interesting in the case of methane. The oc-tupole moment is the first non-zero term in the multipole expansion of the electro-static interaction for methane molecule, so it is crucial in order to describe properlyits dispersion interactions. Actually, the octupole-octupole interaction is the mainlong-range orientation dependent interaction in methane, and what is more, disper-sion energy terms are fundamental in the binding energy of materials containingtetrahedral molecules such as methane. Moreover, the complex charge distributionof methane, which has long been studied in the literature [32, 24, 28], is mainlydependent on its octupole moment, thus, the octupole moment is essential to char-acterize the charge distribution of tetrahedral molecules.For tetrahedral molecules the octupole moment is simply given by one component,namely Ω = Ωxyz. Employing PNOF6(1) with the Sadlej-pVTZ basis set at the ex-perimental equilibrium geometry [28], the result obtained for CH4 is Ωxyz = 2.06a.u.,whereas the experimental mark reported in Ref. [8] is Ωxyz = 2.95 ± 0.17 a.u.Although the PNOF6(1) result falls out of the experimental interval error, theagreement between both values is reasonable taking into account the discrepan-cies between experimental marks obtained by different experimental techniques [8],which shows differences of 1.26 a.u.. Besides, comparing to theoretical calcula-tions, the PNOF6(1) value is very close to the result obtained by using CCSD,Ωxyz = 2.1255 a.u., and to the full-CI value of Ωxyz = 2.005 a.u. obtained by Coe etal. [32] using the cc-pVDZ basis set. Consequently, we can conclude that PNOF6(1)

29


describes properly the Td symmetry of CH4.

30

4 ConclusionsReduced density matrix functional theory is a promising approach to the problemof electron correlation based on the existence of a functional of the one-electronreduced density matrix. One of the possible strategies towards development of novel1-RDM functionals consists of assuming the cumulant expansion for the 2-RDMand approximate the cumulant part, by imposing known and necessary functionaland thereby 2-RDM N−representability conditions. This reconstruction method iscalled the Bottom− Up method.The PNOF6 method, in its extended version, has been assessed by comparing themolecular electric moments with the experimental data as well as with CCSD andMRSD-CI theoretical values. The dipole, quadrupole and octupole moments for aselected set of well-characterized 21 molecules have been calculated at the experi-mental equilibrium geometries using the triple-ζ Gaussian basis set with polarizationfunctions developed by Sadlej. Our results show that PNOF6(Nc) is able to predictelectric properties as accurate as high-level electronic structure methods such asCCSD or MRSD-CI.For PNOF6(Nc) dipole moments, the obtained MAE with respect to experimentaldata is 0.0289 a.u., being consistent with the theoretical benchmark calculations.Remarkable is the result obtained by PNOF6(5) for Carbon monoxide, for which,HF gives a wrong direction of the dipole and CCSD gives a value that overestimatesthe experimental mark severely, whereas PNOF6(5) corrects the sign, giving a resultthat is in excellent agreement with the experimental value.The high performance of PNOF6(Nc) in computing electric quadrupole momentshas been shown by most of the studied molecules, for which the computed valuesfall into the experimental interval error. It has been shown that the method iscapable of providing the different components of the quadrupole moment tensor.The PNOF6(Nc) MAE with respect to the experiment is 0.1194 a.u., which is veryclose to the corresponding MAEs of 0.0902 a.u. and 0.1448 a.u. obtained by usingthe well-established CCSD and MRSD-CI methods, respectively. In particular, theresults obtained for the ozone molecule with a marked multiconfigurational char-acter, show that the method is able to treat properly non-dynamic and dynamicelectron correlations. Also, the result obtained for the well known case of benzeneis excellent. In fact, while benzene does not have a dipole moment, it has a strongquadrupole moment which is essential to determine the preferred geometries of thebenzene dimer, a prototypical system for the study of π−stacking1.

1In chemistry, π−stacking refers to attractive, noncovalent interactions between aromatic rings,

31

Chapter 4 Conclusions

Finally, the study of the octupole moment was focused here in the methane, due toits important role in the description of dispersion interactions of this molecule and itscompounds. The tetrahedral distribution of CH4 was well described by PNOF6(1).In conclusion, the present master thesis proves that PNOF6(Nc) recovers a greatamount of dynamical correlation, as CCSD does, and it has proved to be the mostaccurate method studied in this work when different components of the multipolemoment tensors are studied, what is related to the static electron correlation. Hence,PNOF6(Nc) computes accurately the charge distribution of molecular systems, sothat not only the correlation energies of molecules can be obtained but in additionalso the correlated one-particle reduced density matrix, which, in the context of one-electron reduced density matrix functional theory (1-RDMFT), plays a role analogueto the wave function in ab− initio approaches or the density ρ in DFT.

F UT URE W ORK

Regarding the future work related to the task developed in this thesis, it wouldbe interesting to investigate the intermolecular forces by employing the multipoleelectric moments calculated by using PNOF6(Nc). As we have explained throughthis work, the electric moments are specially useful to compute electrostatic interac-tions, for instance, the quadrupole-quadrupole interaction in benzene dimer, or theoctupole-octupole interaction in methane dimer. In the manner shown in section1.2, dedicated to the definition of molecular electrostatic moments, we can carryout a Taylor expansion of the electrostatic interaction, so that the Hamiltoniancorresponding to a molecular system in an electric field can be cast as

H = H(0) − µαFα −13ΘαβFαβ − ...

where H(0) is that for the free molecule, µα and Θαβ the usual dipole and quadrupolemoments, and Fαβ and Fαβ the electric field and the field gradient at the origin dueto the external charges, and so on. Then, if the molecule is in the internal quantumstate ψ, its energy for a fixed position and orientation is

W = ⟨ψ|H|ψ⟩ = W (0) − µ(0)α Fα − 1

2ααβFαFβ

−16βαβγFαFβFγ − ...−

13Θ(0)

αβFαβ − 13Aγ,αβFγFαβ − ...

where µ(0)α = ⟨ψ(0)|µα|ψ(0)⟩ and Θ(0)

αβ = ⟨ψ(0)|Θαβ|ψ(0)⟩ are the permanent dipoleand quadrupole moments of the molecule, ψ(0) being the unperturbed wave function

since they contain π bonds. Further, these interactions play an important role in nucleobasestacking within DNA and RNA molecules, protein folding, template-directed synthesis, materialscience, and molecular recognition.

32

Conclusions

(i.e., H(0)ψ(0) = W (0)ψ(0)); ααβ, βαβγ, Aγ,αβ, and derivatives are the molecular po-larizabilities describing the distorsion of the molecule by the external electric fieldand field gradient. Essentially, it is possible to calculate the electrostatic energyemploying this expression, just truncating the given formula up to the needed ac-curacy. Thus, the intermolecular forces responsible for the interesting cases such asthe mentioned some rows above can be calculated by using the molecular electricmoments, which have been studied in depth in the present master thesis.Conversely, we can calculate the polarizabilities from the knowledge of molecu-lar energy, and use the obtained results to test and validate the quantum chem-istry method employed to compute that energy. For example, the total dipole andquadrupole moments of the molecule are

µα = −∂W∂Fα

= µ(0)α + ααβFβ + 1

2βαβγFβFγ + 16γαβγδFδFβFγ + 1

3Aα,βγFβγ + ...

Θαβ = −3 ∂W∂Fαβ

= Θ(0)αβ + Aγ,αβFγ + 1

2Bγδ,αβFγFδ + ...

.

Although these magnitudes have been defined before, let us recall that the secondrank tensor α is the static polarizability, and γ and β are hyperpolarizabilitiesdescribing deviations from a linear polarization law. Tensor A determines the dipoleinduced by a field gradient and the quadrupole induced by a uniform field, and soon for the other magnitudes. In other words, these magnitudes allow to know how isperturbed the molecular wave function when an electrical field is applied, and thatis why they play an important role in atomic and molecular physics.

33

Acknowledgments

The autor thanks for technical and human support provided by IZO-SGI SGIker ofUPV/EHU and European funding (ERDF and ESF). Financial support comes froma PhD. grant of the Vice-Rectory for research of the UPV/EHU.Eskerrik asko Mario, hau posible egitearren, eta orokorrean Kimika Teorikoko kideguztiei zuen laguntza zientifikoa eta ez-zientifikoagatik!

35

Bibliography

[1] A. Halkier, H. Larsen, J. Olsen, P. Jørgensen and J. Gauss. Full configurationinteraction benchmark calculations of first-order one-electron properties of BHand HF. J. Chem. Phys., 110:734–740, 1999.

[2] A.F.C. Arapiraca and J.R. Mohallem. DFT vibrationally averaged isotopicdipole moments of propane, propyne and water isotopologues. Chem. Phys.Lett., 609:123–128, 2014.

[3] A. D. Buckingham. Direct method of measuring molecular quadrupole mo-ments. J. Chem. Phys., 30:1580, 1959.

[4] A. D. Buckingham. Permanent and induced molecular moments and long-rangeintermolecular forces. Adv. Chem. Phys., 12:107–142, 1967.

[5] E. Cances and K. Pernal. Projected gradient algorithms for hartree-fock anddensity matrix functional theory calculations. J. Chem. Phys., 128:134108–134116, 2008.

[6] J. Cioslowski, M. Piris, and E. Matito. Robust Validation Of Approximate1-Matrix Functionals With Few-Electron Harmonium Atoms. J. Chem. Phys.,143:214101, 2015.

[7] A. J. Cohen and Y. Tantirungrotechai. Molecular electric properties: an assess-ment of recently developed functionals. Chemical Physics Letters, 299(5):465–472, 1999.

[8] E. R. Cohen and G. Birnbaum. Far infrared collisioninduced absorption ingaseous methane. ii. determination of the octupole and hexadecapole moments.J. Chem. Phys., 62:3807–3812, 1975.

[9] A. J. Coleman. Structure of Fermion Density Matrices. Rev. Mod. Phys.,35:668–687, 1963.

[10] D. J. Gearhart, J. F. Harrison, K. L. C. Hunt. Molecular Quadrupole Momentsof HCCH, FCCF, and ClCCCl. Int. J. Quantum Chem., 95:697–705, 2003.

[11] T. H. Dunning Jr. Gaussian basis sets for use in correlated molecular cal-culations. I. The atoms boron through neon and hydrogen. J. Chem. Phys.,90:1007–1023, 1989.

[12] R. M. Erdahl. Representability. Int. J. Quantum Chem., 13:697–718, 1978.[13] B. Fabricant and J. S. Muenter. Molecular beam Zeeman effect and dipole

moment sign of ClF. J. Chem. Phys., 66:5274–5277, 1977.

37

Bibliography

[14] W. H. Flygare and R. C. Benson. The molecular zeeman effect in diamag-netic molecules and the determination of molecular magnetic moments (g val-ues), magnetic susceptibilities, and molecular quadrupole moments. Mol. Phys.,20:225, 1971.

[15] G. Frenking, C. Loschen, A. Krapp, S. Fau and S. H. Strauss. Electronic Struc-ture of CO - An Exercise in Modern Chemical Bonding Theory. J. Comput.Chem., 28:117–126, 2006.

[16] E. Sicilia G. Luca, N. Russo and M. Toscano. Molecular quadrupole moments,second moments, and diamagnetic susceptibilities evaluated using the general-ized gradient approximation in the framework of Gaussian density functionalmethod. J. Chem. Phys., 105:3206–3210, 1996.

[17] C. Garrod and J. K. Percus. Reduction of the n-particle variational problem.J. Math. Phys., 5:1756–1776, 1964.

[18] T. L. Gilbert. Hohenberg-Kohn theorem for nonlocal external potentials. Phys.Rev. B, 12:2111–2120, 1975.

[19] R. Glaser, Z. Wu, and M. Lewis. A higher level ab initio quantum-mechanicalstudy of the quadrupole moment tensor components of carbon dioxide. J. Mol.Struct., 556:131–141, 2000.

[20] C. J. Goedecker, S.; Umrigar. Theory and Applications of Computational Chem-istry. Springer US, 2000.

[21] Mark S. Gordon and Michael W. Schmidt. Theory and Applications of Com-putational Chemistry. Elsevier, 2005.

[22] W. Gordy and R. L. Cook. Microwave Molecular Spectra. Wiley, New York,1984.

[23] M. J. Gunning and R. E. Raab. Physical implications of the use of primitiveand traceless electric quadrupole moments. Mol. Phys., pages 589–595.

[24] H. N. W. Lekkerkerker, P. Coulon and R. Luyckx. Dispersion Forces inMéthane. J. Chem. Soc., Faraday Trans., 73(2):1328–1334, 1977.

[25] G. L. Heard and R. J. Boyd. Density functional theory studies of the quadrupolemoments of benzene and naphthalene. Chem. Phys. Lett., 277:252–256, 1997.

[26] J. M. Herbert and J. E. Harriman. N-representability and variational stabilityin natural orbital functional theory. J. Chem. Phys., 24:10835–10846, 2003.

[27] A. L. Hickey and C. N. Rowley. Benchmarking Quantum Chemical Methodsfor the Calculation of Molecular Dipole Moments and Polarizabilities. J. Phys.Chem. A, 118:3678–3687, 2014.

[28] E. Hirota. Anharmonic potential function and equilibrium structure of methane.J. Mol. Struct., 77:213–221, 1979.

38

Bibliography

[29] S. Hofinger and M. Wendland. Method/Basis Set Dependence of the TracelessQuadrupole Moment Calculation for N2, CO2, SO2, HCl, CO, NH3, PH3, HF,and H2O. Int. J. Quantum Chem., 86:199–217, 2002.

[30] P. Hohenberg andW. Kohn. Inhomogeneous Electron Gas. Phys. Rev., 136:864–871, 1964.

[31] Russel D. Johnson III, editor. NIST Computational Chemistry Comparison andBenchmark Database, volume 17b. 2015.

[32] D. J. Taylor J. P. Coe and M. J. Paterson. Monte Carlo configuration interactionapplied to multipole moments, ionisation energies and electron affinities. J.Comput. Chem., 34:1083, 2013.

[33] J. D. Jackson. Classical Electrodynamics. New York, NY: Wiley, 3rd ed. edition,1999.

[34] J. M. Junquera-Hernández, J. Sánchez-Marín, and D. Maynau. Molecular elec-tric quadrupole moments calculated with matrix dressed SDCI. Chem. Phys.Lett., 359:343–348, 2002.

[35] T. Helgaker P. Jørgensen K. L. Bak, J. Gauss and J. Olsen. The accuracy ofmolecular dipole moments in standard electronic structure calculations. Chem.Phys. Lett., 319(5-6):563–568, 2000.

[36] A. A. H. Padua L. P. N. Rebelo K. Shimizu, M. F. C. Gomes and J. N. C. Lopes.On the role of the dipole and quadrupole moments of aromatic compounds inthe solvation by ionic liquids. J. Phys. Chem. B, 113(29):9894–9900, 2009.

[37] S. G. Kukolich. Molecular Beam Measurement of the Magnetic SusceptibilityAnisotropies and Molecular Quadrupole Moment in H2CO. J. Chem. Phys.,54:8–11, 1971.

[38] W. Kutzelnigg and D. Mukherjee. Cumulant expansion of the reduced densitymatrices. J. Chem. Phys., 110:2800–2809, 1999.

[39] M. Levy. Universal variational functionals of electron densities, first-order den-sity matrices, and natural spin-orbitals and solution of the v-representabilityproblem. Proc.Natl. Acad. Sci. USA, 76:6062–6065, 1979.

[40] X. Lopez, M. Piris, F. Ruipérez, and J. M. Ugalde. Performance of PNOF6 forHydrogen Abstraction Reactions. J. Phys. Chem. A, 119(27):6981–6988, 2015.

[41] Per-Olov Lowdin. Quantum Theory of Many-Particle Systems. I. Physical In-terpretations by Means of Density Matrices, Natural Spin-Orbitals, and Con-vergence Problems in the Method of Configuration Interaction. Phys. Rev.,97:1474–1489, 1954.

[42] N. H. March M. Piris. Low-lying Isomers of Free-space Halogen Clusters withTetrahedral and Octahedral Symmetry in Relation to Stable Molecules Such asSF6. Phys. Chem. A, 119:10190–10194, 2015.

39

Bibliography

[43] G. Maroulis. On the Electric Multipole Moments of Carbon Monoxide. Z.Naturforsch, 47a:480–484, 1992.

[44] G. Maroulis. Accurate electric dipole and quadrupole moment, dipole polariz-ability, and first and second hyperpolarizability of ozone from coupled clustercalculations. J. Chem. Phys., 101:4949–4955, 1994.

[45] J. M. Matxain, M. Piris, F. Ruipérez, X. Lopez, and J. M. Ugalde. Homolyticmolecular dissociation in natural orbital functional theory. Phys. Chem. Chem.Phys., 13:20129–20135, 2011.

[46] D. A. Mazziotti. Approximate solution for electron correlation through the useof Schwinger probes. Chem. Phys. Lett., 289:419–427, 1998.

[47] D. A. Mazziotti. Structure of fermionic density matrices: Complete n-representability conditions. Phys. Rev. Lett., 108:263002–263007, 2012.

[48] A. L. McClellan. Tables of Experimental Dipole Moments, volume 3. Rahara,El Cerrito, CA, 1989.

[49] J. M. Mercero, J. M. Matxain, X. Lopez, D. M. York, A. Largo, L. A. Eriksson,and J. M. Ugalde. Theoretical methods that help understanding the structureand reactivity of gas phase ions. International Journal of Mass Spectrometry,240(1):37–99, 2005.

[50] G. Meyer and J. P. Toennies. Moments inelastic of light hydrocarbon state tostate. Chem. Phys. Lett., 52:39–46, 1980.

[51] L. J. Morrison R. C., Barolotti. Exchange-correlation potentials for high-electron-density ions in the Be isoelectronic series. J. Chem. Phys.,121(24):12151–12157, 2004.

[52] R. Nelson, D. Lide, and A. Maryott. Dipole Moment Table, 1967.[53] R. H. Orcutt. Influence of Molecular Quadrupole Moments on the Second Virial

Coefficient. J. Chem. Phys., 39:605, 1963.[54] P. Bundgen, F. Grein and A. J. Thakkar. Dipole and quadrupole moments

of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wave functions. J. Mol. Struct, 334:7–13,1994.

[55] W. Parr, R.G.; Yang. Theory and Applications of Computational Chemistry.Oxford University Press, USA, 1994, 1989.

[56] K. Pernal and K. J. H. Giesbertz. Reduced Density Matrix Functional Theory(RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT). TopCurr Chem, 368:125–184, 2016), (and references therein.

[57] M. Piris. A generalized self-consistent-field procedure in the improved BCStheory. J. Math. Chem., 25:47–54, 1999.

[58] M. Piris. A New Approach for the Two-Electron Cumulant in Natural Orbital.Int. J. Quantum Chem., 106:1093–1104, 2006.

40

Bibliography

[59] M. Piris. Natural Orbital Functional Theory. In D A Mazziotti, editor,Reduced-Density-Matrix Mechanics: with applications to many-electron atomsand molecules, chapter 14, pages 387–427. John Wiley and Sons, Hoboken, NewJersey, USA, 2007.

[60] M. Piris. A natural orbital functional based on an explicit approach of thetwo-electron cumulant. Int. J. Quantum Chem., 113:620–630, 2013.

[61] M. Piris. Interacting pairs in natural orbital functional theory. J. Chem. Phys.,141:44107, 2014.

[62] M. Piris, X. Lopez, F. Ruipérez, J. M. Matxain, and J. M. Ugalde. A naturalorbital functional for multiconfigurational states. J. Chem. Phys., 134:164102,2011.

[63] M. Piris, X. Lopez, and J. M. Ugalde. The Bond Order of C 2 from an StrictlyN-Representable Natural Orbital Energy Functional Perspective. Chemistry -A European Journal, DOI: 10.1002/chem.201504491,, 22, 2016).

[64] M. Piris and N. H. March. Is the Hartree-Fock prediction that the chemicalpotential µ of non-relativistic neutral atoms is equal to minus the ionizationpotential I sensitive to electron correlation? Physics and Chemistry of Liquids,2015.

[65] M. Piris and N. H. March. Low-Lying Isomers Of Free-Space Halogen ClustersWith Tetrahedral And Octahedral Symmetry In Relation To Stable MoleculesSuch As SF6. J. Phys. Chem. A, 119(40):10190–10194, 2015.

[66] M. Piris, J. M. Matxain, and X. Lopez. The intrapair electron correlation innatural orbital functional theory. J. Chem. Phys., 139(23):234109–9, 2013.

[67] M. Piris, J. M. Matxain, X. Lopez, and J. M. Ugalde. Spin conserving naturalorbital functional theory. J. Chem. Phys., 131:021102, 2009.

[68] M. Piris, J. M. Matxain, X. Lopez, and J. M. Ugalde. Communication: Therole of the positivity N-representability conditions in natural orbital functionaltheory. J. Chem. Phys., 133:111101, 2010.

[69] M. Piris, J. M. Matxain, X. Lopez, and J. M. Ugalde. The one-electron picturein the Piris natural orbital functional 5 (PNOF5). Theor. Chem. Acc., 132:1298,2013.

[70] M. Piris and J. M. Ugalde. Iterative Diagonalization for Orbital Optimizationin Natural Orbital Functional Theory. J. Comput. Chem., 30:2078–2086, 2009.

[71] E. Ramos-Cordoba, X. Lopez, M. Piris, and E. Matito. H4: A challenging sys-tem for natural orbital functional approximations. J. Chem. Phys., 143:164112,2015.

[72] M. Rosina. Historical Introduction. In D A Mazziotti, editor, Reduced-Density-Matrix Mechanics: with applications to many-electron atoms and molecules,chapter 2, pages 11–18. John Wiley and Sons, Hoboken, New Jersey, USA,2007.

41

Chapter 4 Bibliography

[73] F. Ruipérez, M. Piris, J. M. Ugalde, and J. M. Matxain. The natural orbitalfunctional theory of the bonding in Cr(2), Mo(2) and W(2). Phys. Chem. Chem.Phys., 15:2055–2062, 2013.

[74] A. J. Russel and M. A. Spackman. An ab initio study of vibrational correctionsto the electric properties of the fluoromethanes: CH3F, CH2F2, CHF3 andCF4. Mol. Phys., 98:633–642, 2000.

[75] A.J. Sadlej. Collect. Czech. Chem. Commun., 53:1995, 1988.[76] A.J. Sadlej. Medium-size polarized basis sets for high-level-correlated calcula-

tions of molecular electric properties. Theor. Chim. Acta, 79:123–140, 1991.[77] M. W. Schmidt, . K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H.

Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. U. Shyjun, M. Dupuis,and J. A. Montgomery. General Atomic and Molecular Electronic StructureSystem. J. Comput. Chem., 14:1347–1363, 1993.

[78] J. D. Watts and R. J. Bartlett. Coupled-cluster calculations of structure andvibrational frequencies of ozone: Are triple excitations enough? J. Chem.Phys., 108(6):2511–2514, 1998.

42

5 Appendix I

In this appendix we explain how to program the electric dipole, quadrupole, andoctupole moments in the context of Natural Orbital Functional Theory.

For the nuclear contribution the point-charge approximation has been considered,so the term is simply given by the product of nuclear coordinates and their corre-sponding Coulombic charges. The situation is much more difficult in the case of thenegative charge. This is given by the electronic density, which in NOF theory canbe computed from the multiplication between the square of the module of NaturalOrbitals and their corresponding occupation numbers, i.e

ρNO (r) =∑i

ni|φNOi |2. (5.1)

Thus, in the basis of Natural Orbitals the electric dipole moment is computed by

−→µe = ⟨ρ · (x, y, z) ⟩ = ⟨ψ0| −∑i

ri|ψ0⟩ = −2∑i

ni⟨φNOi |ri|φNOi ⟩, (5.2)

where a factor 2 comes from considering a singlet state system. In practice, we workin the Atomic Orbital basis, so after expanding the Natural Orbitals in AtomicOrbitals, φNOi = ∑

k CkiχAOk , the expression of the dipole moment is

−→µe = −2∑i

ni∑k,l

CkiC∗il⟨χAOl |ri|χAOk ⟩, (5.3)

where the set of integrals ⟨χAOk |ri|χAOl ⟩, known as 1st-order moment integrals, hasbeen directly read from the GAMESS program [77, 21]. Note that the first summa-tion runs over the Natural Orbital basis, whereas indices k and l run over the basisof Atomic Orbitals.

The procedure is completely analogous for higher multipole moments, so only thelast expressions of quadrupole and octupole moments are reported in the next lines.

43

Chapter 5 Appendix I

In the case of quadrupole moment, the traceless formula introduced by Buckinghamleads to the expression

Θi,j = −∑k

∑l,t

nkClkC∗kt

[3⟨χAOt |rirj|χAOl ⟩− ⟨χAOt |r2

i δij|χAOl ⟩], (5.4)

where the factor 2 is compesated with a fraction 12 that is implicit in the definition

of the quadrupole moment. For instance, two relevant cases are

Θx,x = −∑k

∑l,t

nkClkC∗kt

[2⟨χAOt |rxrx|χAOl ⟩− ⟨χAOt |ryry|χAOl ⟩− ⟨χAOt |rzrz|χAOl ⟩

],

and

Θx,y = −∑k

∑l,t

nkClkC∗kt

[3⟨χAOt |rxry|χAOl ⟩

].

For the octupole moment, the formula leads to the next expression for programming:

Ωi,j,k = −∑m

∑l,t

nmClmC∗mt

[5⟨χAOt |rirjrk|χAOl ⟩− ⟨χAOt |r2

i (riαδβγ + riβδαγ + riγδαβ) |χAOl ⟩]

(5.5)

Note that the factor 2 arising from doubly occupancies is removed again. In thiscase the relevant cases are three:

Ωx,x,x = −∑m

∑l,t

nmClmC∗mt

[2⟨χAOt |rxrxrx|χAOl ⟩− 3⟨χAOt | (rxryry + rxrzrz) |χAOl ⟩

],

Ωx,x,y = −∑m

∑l,t

nmClmC∗mt

[4⟨χAOt |rxrxry|χAOl ⟩− ⟨χAOt | (ryryry + ryrzrz) |χAOl ⟩

],

and

Ωx,y,z = −∑m

∑l,t

nmClmC∗mt

[5⟨χAOt |rxryrz|χAOl ⟩

].

Recall that both quadrupole and octupole tensors are symmetric in all suffixes.

44

Molecular Electric Moments Calculated By Using Natural Orbital Functional Theory · 2016. 5....

Documents

Transcript of Molecular Electric Moments Calculated By Using Natural Orbital Functional Theory · 2016. 5....