Università di Pisa
DIPARTIMENTO DI FISICA
Corso di laurea Magistrale in Fisica
Analysis of nondynamic electron
correlation in DFT
Candidato
Carmelo Naim
Relatori
Prof. Claudio Amovilli
Dr. Eduard Matito
Acknowledgments
Vorrei ringraziare in primo luogo il professor Amovilli per avermi dato tutti gli strumenti utili per la realizzazzione di questa tesi, ma anche per essere stato paziente e disponibile ogni qualvolta avessi bisogno di aiuto.
A special thanks goes to dr. Eduard Matito who gave me the opportu-nity to work with him and his group at DIPC, it was a very formative and beautiful experience. I am grateful also to dr. Eloy Ramos-Cordoba for his guidance during the realization of this work and the other people of the group who followed me during my time in Spain: dr. Mauricio, Sebastian, mama Mireia, Irene and Xiang.
La realizzazione di questa tesi è stata possibile soprattutto grazie alle persone che mi hanno sostenuto durante il mio percorso d studi, in partico-lare la mia famiglia per avermi dato la possibilità di studiare a Pisa e avermi supportato anche nei momenti di cili. Ringrazio i miei genitori che sono la mia fonte di ispirazione e mi hanno sostenuto sia economicamente che per-sonalmente durante questi anni ed Erica sulla quale nonostante le litigate so di poter sempre contare.
Ringrazio Allegra che negli ultimi sei anni mi è stata accanto nonostante la lontananza e tutte le di coltà, senza il suo sostegno non sarei mai arrivato n dove sono ora.
Vorrei ringraziare anche i colleghi con i quali ho condiviso questi cinque anni di corso. In particolare Laura, Eugenia e Ziparo che mi sopportano dal primo giorno di università e sono sempre stati presenti negli anni successivi. Luca per le mille avventure vissute assieme, Meri e Nicola per l'aiuto e le risate degli ultimi anni.
Ringrazio inoltre tutte le altre persone che hanno reso speciali questi anni a Pisa, in particolare i membri della FAIGGHIA con i quali ho condiviso
bel-lissime esperienze: Sergio, Robi,Giammarco, Dario e Sopie.
Per ultimi ringrazio tutte le persone che da Reggio mi hanno sostenuto e reso speciali le mie vacanze da fuorisede. In particolare ringrazio Franca per essere stata sempre presente quando ne avevo bisogno, Alberto che sin dal liceo mi fa innervosire e tutti i ragazzi della coop che da sempre mi stanno vicini: Dalila, Stefano, Elisa, Arturo, Jacopo, Paolo, Fabio e Daniele.
Contents
Introduction 3
1 Electron Correlation 6
1.1 Dierent types of electron correlation . . . 7
1.1.1 Unrestricted framework . . . 8
1.2 Post HF wavefunction methods . . . 8
1.2.1 HF based methods . . . 9
1.2.2 Multicon gurations theories . . . 11
1.3 Quantum Monte Carlo Methods . . . 13
1.3.1 Stochastic integration . . . 13
1.3.2 Sampling of the con gurations . . . 14
1.3.3 Calculation of energy . . . 15
1.3.4 De ne the QMC wavefunction . . . 15
1.4 Reduced Density matrix formalism . . . 17
1.4.1 Natural orbitals . . . 18
1.5 Density Functional Theory . . . 19
1.6 Indicators of dynamic and nondynamic correlation . . . 21
2 Developments in DFT 24 2.1 Density functional approximations . . . 24
2.1.1 DFT problems and limitations . . . 27
2.1.2 Electron correlation in DFT . . . 28
2.2 DFT beyond KS approach . . . 29
3 Fit of a Correlated density matrix 30 3.1 Reference density with QMC approach . . . 30
3.1.1 Correlated 1-RDM . . . 31
3.1.2 Pair density from the 1-RDM . . . 33
3.2 The tting procedure . . . 34
3.3 Test cases . . . 34
3.4.1 Electronic density . . . 36 3.5 Fitting of the density . . . 36 3.6 Indicators of nondynamic correlation . . . 40
4 Broken Symmetry occupancies for DFT calculation 44
4.1 Broken symmetry solution . . . 44 4.2 Tested functionals . . . 46 4.3 Results of the calculations . . . 48
4.3.1 Indicators of nondynamic and dynamic correlation from Broken Symmetry occupancies . . . 50
5 Fractional occupancies obtained from RDFT energies 56
5.1 Fit of the QMC occupancies with KS energies . . . 57 5.1.1 Results for one parameter . . . 58 5.1.2 Results for two parameters tting . . . 58 5.1.3 Results for two parameters with localized orbitals
en-ergies . . . 69 5.2 Broken symmetry occupation numbers for DFT calculation . . 69 5.2.1 Fit for canonical orbitals . . . 69 5.2.2 Localized orbitals . . . 80 5.3 Local indicators of nondynamic correlation . . . 88
Introduction
One of the main goals of quantum chemistry is the accurate calculation of the ground state energy of molecules and solids. A basic step in this kind of com-putation is the analysis of the electronic structure by solving the Schrödinger equation in the Born Oppenheimer approximation.
Nowadays, Density Functional Theory (DFT) is one of the most used methodologies for solving the electrons structure problem. This method is preferred to others because it balances a good accuracy with a low com-putational cost. The strength of DFT is that this method is based on the determination of a density function instead of a N-particle wavefunction. In the Kohn Sham approach of DFT, the problem of nding the real system density is mapped to a system of non-interacting electrons in an eective potential. DFT is formally exact, however, an explicit expression of the exchange and correlation function, is still unknown. Therefore, the develop-ment of DFT passes through the nding of such an accurate exchange and correlation functional. Several Density functional approximations (DFAs) have been developed since this method has been proposed, Perdew collected them in the so-called Jacob's ladder, ordering the functionals according to the complexity and the accuracy obtained.
Despite the undeniable success of DFT is in a continuous growing there are still some problems which are far from being solved. In particular, in this work, we analyze the de ciency of DFT in describing nondynamic cor-relation. Nondynamic correlation is important for systems in which two or more con gurations are requested for the correct description of the electronic structure, like for example diradical systems or homolytic dissociations. Typ-ically, a multideterminant wavefunction is needed to achieve numerical ac-curacy. In quantum chemistry, several methods which take into account nondynamic correlation have been developed. Worthy of mention, for this kind of problems, are the methods Complete Active Space Self Consistent Field(CASSCF) and Generalized Valence Bond(GVB). In this thesis work,
we go beyond these methods by using a QMC approach.
In DFT, until now, there is not a satisfactory functional approximation which gives a comprehensive description of nondynamic correlation. In order to develop new kind of functionals where the eects of nondynamic correla-tion are accounted, a clear identi cacorrela-tion of these eects is mandatory. The analysis of the correlation eects has been done for wavefunction methods through correlation indicators based on natural orbital occupancies. How-ever, KS-DFT, being based on an independent particle model, prevents the use of these indicators. In this work, we extract information on fractional occupancies following two dierent routes. The calculations have been per-formed in systems in which nondynamic correlation is important: ortho-benzyne, meta-ortho-benzyne, and para-benzyne. These molecules have a biradical character.
The rst approach consists in obtaining accurate occupation numbers from a correlated electronic density generated in a QMC computation. We realized a t of the correlated QMC density through a GVB expansion of DFT orbitals, and the occupancies have been obtained from the tted pa-rameters. We checked the accuracy of the resulting occupation numbers obtained through a comparison between the accurate kinetic energy of QMC and the corresponding GVB one.
In the second analysis, we used an unrestricted DFT calculation. This technique allows to approximate, better than Restricted DFT, the ground state energy of multicon gurational systems, however, they bring spin con-tamination in the KS determinant. Through the projection of the KS density matrix on a natural orbitals basis set, it is possible to extract fractional oc-cupancies. This calculation has been performed for several DFAs, showing the dierence of the inclusion of electron correlation for dierent approxima-tions in climbing Jacob's ladder. The results have been compared with the QMC reference, with the aim to understand which functionals reproduce in a better way these eects.
Recent developments of new functionals are based on the extension of the restricted framework with fractional occupation numbers. In order to make a contact with these new methods we have tted our results to a Fermi distribution with restricted KS energies. In this context, we obtained two parameters, namely the eective temperature and the chemical potential, which are able to give back the occupation numbers by the minimization of
Finally, this analysis could be useful for the development of new kinds of functionals based on local decomposition taking account also of nondynamic eects.
Chapter 1
Electron Correlation
The evaluation of potential energy surfaces, namely the function Eel( ~R) of
molecules is a primary goal of computational chemistry. In order to give an accurate description of it, it is necessary to solve the electronic structure problem. If the system is treated by rst principles(Abinitio) methods, we need to solve the many-body, time-independent, non-relativistic Schrödinger equation in the Born-Oppenheimer approximation, which could be written in atomic units as follows:
− Nel X i ∇2 i 2 − 1 2 Nel,N X i,I ZI |RI− ri| +1 2 Nel X i6=j 1 |ri− rj| Ψ = EΨ (1.0.1)
Where Nel is the number of electrons, N is the number of nuclei, ZI the
charges of nuclei, ri the positions of the electrons and RI the positions of
the nuclei. It is not possible to solve Eq. 1.0.1, so the energy can only be obtained as the expectation value of the electronic Hamiltonian above, given an appropriate approximation for the wavefunction. The most limiting factor for this problem is the description of the interaction of electrons which is fundamental for determining the chemistry of the system. The interaction of electrons could be explained by two dierent contributes:
Statistics correlation: the electrons obey to Fermi Statistics, the wave-function has to be antisymmetric with respect to the exchange of the coordinates of two electrons.
Coulomb interaction: the electrons have to interact with the repulsive Coulomb potential (in the Hamiltonian of Eq. 1.0.1).
ing as trial wavefunction the antisymmetrized product of single particles orbitals. This method assures the correct treatment of Fermi Statistics, but the self- consistent approach neglect the instantaneous interaction between electrons, in fact, every electron moves in a mean- eld generated by oth-ers electrons. Near the equilibrium geometries, the HF method guarantees in general, a good initial approximation for the ground state energy of a molecule ( 98% of the exact energy); Lowdin [Lowdin, 1959] de ned the correlation energy as the dierence between exact and HF energies values, namely eq 1.0.2.
Ecorr = Eexact− EHF (1.0.2)
An accurate description of electron correlation is fundamental to treat the properties of molecules such as chemical bonds, geometric structure and phys-ical observables. In order to treat the electron correlation eects a lot of dierent methods had been developed. The electron correlation could be de ned for each method by the Pine de nition in Eq. [Pines, 1953]:
EcorrX = EX − EHF (1.0.3)
Where X is the method used to treat the system. In this way, the electron correlation became an intrinsic quantity of the method of calculation.
1.1 Dierent types of electron correlation
There are dierent ways to classify the electron correlation. A way to clas-sify the electron correlation was given by Sinanoglu [Sinanoglu, 2007], which splits the electron correlation into two parts, dynamic and nondynamic. Dynamic correlation (DC) accounts for the correlation of the movement of the electrons, it comes when the HF framework is consistent but incomplete, in particular, it comes from the di culty of HF to model interelectronic cusps. Indeed the wavefunction, in order to behave properly at short distances, must satisfy the Kato conditions [Kato, 1957]. For this reason, the dynamic corre-lation usually can be associated with a Short-range correcorre-lation(but it is not always true). Nondynamic electron correlation(NDC) arises when the energy is lowered by the interaction of the HF con guration with low-lying con gu-rations, so it comes from the inadequacy of the single determinant framework. Some authors make some distinction between static and nondynamic correla-tion. However, in this work, these terms are going to be treated as synonyms. An explanatory example of the eects of NDC is the di culty of a single determinant picture to describe homolytic bond dissociation. For example,
in the H2 dissociation, if the electrons are far from each other the HF picture is physically unreasonable. In this case, if the distance between the two hydrogen atoms is high, each electron should be on one single nucleus, but this does not happen in HF because in this approximation the electrons have to stay on both nuclei with the same probability. The energy will result
overestimated. The potential energy surface of the H2 molecule is shown in
Fig. 1.1, which shows a comparison between the HF energy and the exact one. It is possible to notice that at short interaction distances, the dynamic correlation is prevalent, but for big distances, NDC is much more important.
Therefore, to give the correct description of H2 dissociation it is necessary
to account for nondynamic correlation, considering interactions of dierent con gurations. The HF approximation is therefore not enough accurate, it is necessary to overcome it. From the next section, an overview of dierent methods beyond HF is shown.
1.1.1 Unrestricted framework
In the previous example, we referred to Restricted HF(RHF). In RHF, elec-trons with dierent spin components are assigned to the same set of orbitals. This constraint is removed in Unrestricted Hartree Fock (UHF). Where a dierent set of orbitals are used for electrons in dierent spin states (α and
β). This brings to a more exible wavefunction. Therefore, the energy would
be lower in principle, than in HF approach, and there will be a more accurate description of the energy at bond dissociation.
The problem of this approach is that the wavefunction does not preserve the
spin symmetry, it is not anymore an eigenfunction of the operator ˆS2(spin
contamination). So it gives a better description of the energy but the physics of the wavefunction is poor. As an example, we could see the potential energy
surface of H2 in Fig. 1.1.
1.2 Post HF wavefunction methods
The HF method is limited by the inadequate description of the wavefunction of the system, in order to improve it with more accurate descriptions dierent
methods have been developed. The limit of the HF wavefunction (ΨHF) is
that, even if it takes into account of Fermi correlation, it does not count in an accurate way of Coulomb interaction; in HF approximation the electrons do not interact through a local instantaneous potential, but through an eective mean eld potential constructed solving the Roothaan equations [Roothaan, 1951].
2 4 6 8 10 12 R[Bohr] 1.1 1.0 0.9 0.8 Energy[Hartree]
PES of H2
HF UHF CISDFigure 1.1: Potential energy curve(PEC) for H2 molecule. Comparison
be-tween CISD, HF and UHF
In order to improve the accuracy in the description of Coulomb interaction, it is necessary to go beyond the single determinant approximation.
In correlated methods, the wavefunction is taken as a linear combination of many electrons con gurations. By starting from an HF con guration it is possible to expand the wavefunction, by adding more con gurations.
1.2.1 HF based methods
In the HF method, the wavefunction ΨHF is characterized by the
antisymmet-ric product of one-electron spin-orbital (Slater determinant wavefunction). The one-electron spin-orbitals are the eigenfunctions of the Fock operator, so if there are n electrons and N orbitals, there will be n occupied spin-orbitals and N − n unoccupied(virtual) ones.
In the HF-based methods, the correlation eects are obtained allowing the wavefunction to be a linear combination of dierent con gurations; the main
term of the wavefunction is the HF one (ΨHF), while the other con gurations
are obtained moving the electrons from the occupied orbitals to the virtual's one. This process is analog of applying excitations operators to the reference wavefunction, the number of electrons that are promoted form occupied to
the virtual spin-orbitals de ne the order of the excitation. The way in which the coe cients of every excitation are evaluated dierentiates the method used.
Perturbative methods The easiest way to improve HF wavefunction is through the perturbation theory. The most common and widely used per-turbation approach is the Mollet Plesset perper-turbation theory [Cremer, 2011], which consists in taking the Fock operator as 0th order, and then correcting perturbatively the wavefunction.
Since the B-O Hamiltonian contains only one and two electron operators, only the single and double excitation will contribute in the expression of the wavefunction. However, the Brillouin theorem prevents the direct mixing
between the single excitations and the ΨHF term, so the second and third
order energies correction have the contribution of just the double excitations. The wavefunction is then corrected from the rst order, meanwhile, the cor-relation energy correction starts from the second order. The perturbation theories are particularly appropriate for taking account of dynamic correla-tion.
Con guration interaction The most general way to perform a multi-determinant calculation is the Con guration Interaction method (CI), that consists in taking a wavefunction which is a linear combination of Slater De-terminants. Every con guration is obtained by promoting a given number of electrons from occupied orbitals generated in the HF framework to vir-tual ones. The coe cients of the expansion are calculated by minimizing the energy. If it contains all the possible excitations this method gives the exact solution with a given basis set. This procedure is called Full-CI ap-proach(FCI) and the form of the wavefunction is the one in the equation 1.2.1.
ΨF CI = N det X i ciΨi = C0ΨHF + N single X k CkSΨ(S)k + N double X j CjDΨ(D)j + ... (1.2.1)
This method is the most accurate possible, but it is also very expansive so it is in practice usable only for small molecules with few atoms. In order to reduce the cost of the calculation, it is possible to truncate the FCI expansion. This truncation could introduce some problems in the calculation such as the size-consistency problem.
Coupled-Cluster The Couple-Cluster(CC) theory consists in assuming
that the CC wavefunction Ψcc will be the eigenstate of the Hamiltonian of
the system [Paldus, 2005]: ˆ
HΨcc= EccΨcc (1.2.2)
This wavefunction is the HF one, to which has been applied an exponential
operator eTˆ , as in eq. 1.2.3: Ψcc = e ˆ TΨ HF = e ˆ T1+ ˆT2+...Ψ HF (1.2.3)
where the operator ˆT is a sum of all the excitations operators, and it could
be expanded through a Taylor expansion as in eq. 1.2.4.
eTˆ = 1 + ˆT + ˆT2+ ˆT3+ ... = 1 + ˆT1+ ˆT2+ ˆT3+ 1 2 ˆ T12+ 1 2 ˆ T1Tˆ2+ ... (1.2.4)
The peculiarity of this method is that for calculating the wavefunction is not necessary to diagonalize the Hamiltonian of the system, but the calculation is moved in a solution of a system of non-linear algebraic equations which could be solved with iterative solutions. This technique is not variational, so it could be both an upper or a lower bound of the real energy of the system.
1.2.2 Multicon gurations theories
The techniques shown above to treat electron correlation belong to a class of methods which uses the HF determinant as a reference. In these methods new con gurations are generated as relatively small corrections, therefore these methods are good for treating dynamic correlation. However, for systems where the HF description is completely inadequate, we need to use dierent approaches. Usually, these systems show nondynamic correlation problems, therefore they have to be treated with two or more con gurations on equal footing. These kinds of methods are called multicon gurational methods. Multicon gurational Self Consistent Field Method(MCSCF) This method consists in selecting a relatively small number of CI con gurations and optimizing simultaneously the coe cients of the determinants and the orbitals by a variational procedure. The important con gurations are usually chosen by knowing some particular properties of the system under investiga-tion, introducing in this way a bias in the calculation.
In general, the MCSCF is computationally expensive. Typically, the elec-trons are distinguished as active or inactive. Inactive elecelec-trons are assigned
to doubly occupied orbitals, while active electrons are used to obtain the so-called active space, namely, the space of functions spanned by the determi-nants constructed by a given set of orbitals which are occupied in all possible ways by the active electrons. The choice of the active space is not unique. Typically, active electrons are a subset of valence electrons. Such a method is called CASSCF(Complete Active Space Self-Consistent Field) [Siegbahn et al., 1981]. Obviously, if the active space selected contains all the orbitals of the basis set and with all the electrons treated as "active", this treatment is equivalent to the FCI expansion. This method could be taken as a refer-ence for nondynamic correlation, for this reason, Handy and coworkers [Mok et al., 1996] have de ned the NDC as follows.
EN D = ECASSCF − EHF (1.2.5)
In order to take account also for dynamic correlation some perturbative meth-ods of the CAS have been developed, where the perturbation is applied on the CAS reference function, an example is the CASPT2 method [Andersson, 1995].
General Valence Bond method The General Valence Bond(GVB) ap-proach is a multicon gurational method based on VB theory. In this method, the wavefunction is expanded by "localized" double excitation between bond-ing(occupied) and antibonding(unoccupied) orbitals [Goddard et al., 1973]. This coupling between bonding and antibonding orbitals is called perfect pairing approach(PP). Once a reference set of single particle spin-orbitals are chosen, the GVB wavefunction could be expressed as the antisymmetrized product of double particle orbitals. These orbitals are constructed as follows:
˜ ψi(1, 2) = h Caφa(1)φa(2) + Cbφb(1)φb(2) i (α(1)β(2) − α(2)β(1)) (1.2.6)
In eq. 1.2.6 , Ca and Cb are the variational coe cients. If the system has
N electrons the overall GVB wavefunction has the shape of Eq. 1.2.7 which
contains N
2 double particle spin orbitals.
ΨGV B = ˆAh ˜ψ1(1, 2) ˜ψ2(3, 4)... ˜ψN
2(N − 1, N )
i
(1.2.7) This multicon guration character allows the GVB method to include non-dynamic correlation. The eects of non-dynamic correlation can be taken into account by using GVB wavefunction as a starting point for a CI expansion.
1.3 Quantum Monte Carlo Methods
The Quantum Monte Carlo method oers a dierent way of treating the electron correlation.
These techniques are based on random sampling. The simplest version of them is the Variational Monte Carlo (VMC) which uses a stochastic integra-tion method for calculating expectaintegra-tion value for a chosen trial optimized wavefunction.
In this chapter, the VMC method is described, following the approach of [Foulkes et al., 2001].
1.3.1 Stochastic integration
In QMC, the multidimensional integrals are evaluated sampling the function of interest in a space of con gurations generated through a random process. Then an average of the results is done for all the obtained values.
Let us de ne a 3N − dimensional vector ~R which contains the coordinates of all the electrons.
~
R = (~r1, ~r2, ..., ~rN) (1.3.1)
The probability density of nding the system of electrons in the position ~R is P( ~R); and it has the following properties:
P( ~R) ≥ 0 (1.3.2)
Z
d ~RP( ~R) = 1 (1.3.3)
A set of uncorrelated con gurations { ~Rm : m = 1, M } , distribuited
accord-ing to P( ~R) is thus chosen.
We de ne a random variable Z as:
Z = f ( ~R1) + f ( ~R2) + ... + f ( ~RM)
M (1.3.4)
Where f is a probability distribution with mean µf and variance σ2
f.
Be-cause of central limit theorem Z is normally distribuited with mean µf and
standard deviation σf/√M.
We want to calculate an integral I of a particular quantity g( ~R) in the 3N-dimensional space:
I = Z
In order to evaluate it stochastically it is necessary to introduce an impor-tance function P( ~R) and rewrite the integral as:
I = Z
d ~Rf ( ~R)P( ~R) (1.3.6)
Where f( ~R) = g( ~R)
P( ~R). The integral I could be evaluated as a set of random
vectors from the distribution P( ~R) and computing the sample average: I = lim M →∞ 1 M M X m=1 f ( ~Rm) (1.3.7)
Choosing an high number of con guration will give much more accurate re-sults.
1.3.2 Sampling of the con gurations
Stocastic integration involves a sampling from a given probability distribution in an high-dimensional space. The most popular method is the Metropolis
algorithm. This method generates a sequence of sampled points ~Rm by
mov-ing randomly a smov-ingle walker accordmov-ing to the followmov-ing steps: 1. Start at a random position ~Ri
2. Make a trial random move to a new position ~Rf chosen from a
proba-bility density function T ( ~Rf ← ~Ri)
3. accept the trial move with probability
A( ~Rf ← ~Ri) = min 1,
T ( ~Ri ← ~Rf)P( ~Rf)
T ( ~Rf ← ~Ri)P( ~Rf)
!
Where P( ~Rf)is the probability that the system is in the con guration
~
Rf. If the trial move is accepted, the vector ~Rf will be the initial value
of the next step; if it is rejected, the initial value does not change. 4. Return to step (2).
The resulting points will be distributed according to P. If this process is repeated on a big number of walkers, we could say that the average number of walkers in the volume element d ~R is n( ~R)d ~R, where n( ~R) is the equilibrium density of walker. Once the equilibrium is established we assure the detailed balance condition, assuming that the average number of walkers moving from
d ~Ri to d ~Rf is the same as the ones doing the inverse path.
The probability of the move of the walker is d ~RfA( ~Rf ← ~Ri)T ( ~Rf ← ~Rf).
So the average moving from a con guration to another is:
d ~RfA( ~Rf ← ~Ri)T ( ~Rf ← ~Rf) × n( ~Ri)d ~Ri (1.3.8)
This quantity has to be balanced with the number of walkers moving reverse, obtaining the following relation for the equilibrium distribution:
n( ~Ri) n( ~Rf) = A( ~Ri ← ~Rf)T ( ~Ri ← ~Rf) A( ~Rf ← ~Ri)T ( ~Rf ← ~Rf) = P( ~Ri) P( ~Rf) (1.3.9) So the equilibrium density n( ~R) is proportional to P( ~R).
1.3.3 Calculation of energy
The expectation value of the electronic Hamiltonian of the system ˆHe is
evaluated with a trial wavefucntion ΨT, which is an upper bound to the
exact ground state energy. If we write:
EV M C = hψT|H|ψTi hψT|ψTi = R d~rψT(~r) ∗Hψ T(~r) R d~r|ψ(~r)|2 = Z d~rρ(~r)EL(~r) In which ρ(~r) = |ψ(~r)T|2
R d~rψ(~r)T|2 is the normalized probability density. The value
EL = Hψ(~ψ(~r)r) is called local energy.
The variational energy could be calculated valuating the average local energy
on a sample of random M points ~Rk evaluated from the probability density,
namely EV M C ∼ 1 M M X k El(Rk)
1.3.4 De ne the QMC wavefunction
In our QMC calculations, we have chosen the trial wavefunction in a Slater-Jastrow form: a combination of Slater determinants multiplicated by the Jas-trow factor [Filippi and Umrigar, 1996] which takes into account the Coulomb
interactions at short distances. Due to the fact that our Hamiltonian does not contain the spin, we can write
ψT(~r1, ..., ~rN) = J Ndet X k dkDkup(~r1, ..., ~rNup)D down k (~rNup+1, ..., ~rN)
Where the Jastrow factor J has the form in the eq. 1.3.10.
J = JAJBJC (1.3.10)
It is made by three components: Ja takes into account the eects of the
interaction of the nuclei and the electrons(e-n), Jb stands for the electronics
two-body interaction (e-e), meanwhile Jc is for the three-body interaction
between two electrons and a nucleus. Each factor is parametryzed, and the parameters are optimized through a VMC step. The fully expression of these factors are the following.
ln(Ja) = natomi X k N X i A(k)0 Rki 1 + A(K)1 Rai + A(K)2 R2ai+ A(K)3 R3ai+ ... ln(Jb) = N X i<j B0Rij 1 + B1Rij + B2R 2 ij + B3Rij3 + ... ln(Jc) = natomi X k N X i<j Ckij(Rki+ Rkj)
The number B0 will be 1/2 for two electrons with opposite spins, and 1/4
for electrons with the same spin. The local energy is calculated at every sampled con guration. The use of this kind of wavefunction guarantees a high accuracy of the method, with a very good description of both dynamic and nondynamic correlation.
Cusp conditions
In writing the wavefunction, it is necessary to take into account the singu-larity that occurs when two electrons collide due to Coulomb repulsion. In order to remove such singularity, the wavefunction must satisfy appropriate cusp conditions. For the B-O approximation, Kato [Kato, 1957] proved that the cusp condition for the e-e collision is :
h ∂ i
Where γ must be 1/2. For electrons in the same spin state should be 1/4 but in this case the problem of coalescence is reduced by Fermi correlation. For the e-n Coulomb interaction there is an analogue cusp condition which has the following equation:
h ∂ ∂~r1α ψ(~r1, ~r2, ..., ~rn) i ~r1α→0 = −Zαψ(~r1, ~r2, ..., ~rn) (1.3.12)
Where Zα is the charge of the nucleus α, and ~r1α is the distance between an
electron and a nucleus.
The coalescence conditions must be satis ed for all the electrons and the nuclei which participate in the calculation.
1.4 Reduced Density matrix formalism
As we have seen in the previous methods the many electrons wavefunction
Ψ is a complex object which is very di cult to manage.
However, in practical applications, we do not need such a detailed description of the system.
The Many-body problem could be expressed in the Reduced density matrices formalism (RDM). According to this method, any observable is obtainable by an RDM of order p ≤ Nel. Most of the physically useful properties could be evaluated just knowing the rst and the second-order reduced density matrix of the system, like for example energies and dipole moments.
The p-RDM for an N-electron system represents the probability of an
elec-tron to be in a position ~r1 knowing the positions of the other p-1 ones.
Γp(~x1, ..., ~xp; ~x10, ..., ~x 0 p) = N p Z d~xp+1...d~xNψ(~x, ..., ~xN)ψ∗(~x10, ..., ~xN) (1.4.1) In order to compute the energy of the system we need just the 1-RDM and the 2-RDM, namely: Γ1(~x1; ~x10) = N Z d~x2...d~xNψ(~x1, ..., ~xN)ψ∗(~x10, ..., ~xN) (1.4.2) Γ2(~x1~x2; ~x10~x20) = N (N − 1) 2 Z d~x3...d~xNψ(~x1, ..., ~xN)ψ∗(~x10, ..., ~xN) (1.4.3)
The Dirac density matrix γ is de ned as the spin-reduced 1-RDM, and it has the following expression:
γ(~r; ~r0) =
Z
While, the electronic density of the system is de ned as the diagonal part of the γ :
ρ(~r) = γ(~r; ~r) (1.4.5)
Finally the pair density ρ2(~r1, ~r2)is given by spin integration of the 2-RDM:
ρ2(~r1, ~r2) =
Z
dσ1dσ2Γ2(~x1, ~x2; ~x1, ~x2) (1.4.6)
Thus, the ground state energy of the system could be re-written knowing the expression of the quantities de ned above, namely:
E0 = hψ0| ˆHe|ψ0i = − 1 2 Z ∇2γ(~r, ~r0 )|~r=~r0− X alpha Z d~r ρ(~r)Zα |~r − ~Rα|+ Z d~r1d~r2 ρ2(~r1, ~r2) |~r1− ~r1| (1.4.7) The quality of the p-RDM depend on the methods used for the calculation.
In the HF theory Dirac density matrix γ(~r; ~r0) x all other p-RDM.
In particular, the 1-RDM is:
ρ(~r) =X
i
φi(~r)∗φi(~r) (1.4.8)
Meanwhile the two body density will be:
ρ2(~r1, ~r2) = ρ(~r1)ρ(~r2) − |γ(~r1, ~r2)|2 (1.4.9)
So it is dependent just on the 1-RDM.
1.4.1 Natural orbitals
In order to express the properties of a system the natural orbitals(NOs) are important quantities, they have been de ned by Löwdin as the single particle eigenvectors of the the 1-RDM 1.4.10.
(ˆγφi)(~r) =
Z
d~r0γ(~r, ~r0)φi(~r0) = niφi(~r) (1.4.10)
where ni are the occupation numbers which have to satisfy the conditions:
X
i
ni = Nel, 0 ≤ ni ≤ 1
Using eq. 1.4.10, it is possible to decompose the 1-RDM on the basis of NOs as follows:
For the HF system, the natural orbitals correspond to the single particles SCF orbitals, and they have just 0 and 1 as occupation numbers, while, for correlated electrons, in which the wavefunction of the system is a combination of determinants, they have fractional values.
If we consider a multicon gurational method in which the wavefunction is a linear combination of Slater determinants, made of orthogonal orbitals is possible to express it as:
Ψ0 = X k ckΦk(~r) (1.4.12) where Φk is Φk = ˆA[φ1..φN] (1.4.13)
The natural NOs could be expressed as eq. 1.4.14
ni = X
(i|k∈Φk)
c2k (1.4.14)
1.5 Density Functional Theory
The DFT method nowadays is the cheapest and most versatile method of elec-tron structure calculation. The main idea of DFT is to solve the many-body problem, without the construction of an accurate wavefunction, but through a calculation of the electronic density. The consistency of this method is due to Hohenberg and Kohn theorem [Hohenberg and Kohn, 1964], which establishes that there is a one to one correspondence between the external potential acting on the electrons and the electronic density. The consequence of the HK theorem is that the expectation value of the ground state of a many-electron system is a functional of the electron density as follows:
E0[ρ] = T [ρ] + VeN[ρ] + Vee[ρ] (1.5.1)
The problem of calculating the ground state energy is then reduced from the nding of a wavefunction of 3N coordinates to the nding of a function of 3 variables, the electronic density. In order to solve Eq. 1.5.1 above, the basic approach is the Kohn-Sham [Kohn and Sham, 1965], in which a system of non-interacting electrons with the same electronic density of the real system is taken as the reference. For a system of independent particles the density has the following form:
˜
ρ(~r) =X
i
where φi are the single particle orbitals and the sum runs over all the occu-pied orbitals of the reference system.
The expression of the ground state energy is therefore rewritten in terms of the KS electronic density in eq. 1.5.2 as
E0[ ˜ρ] = Ts[ ˜ρ] + VeN[ ˜ρ] + VH[ ˜ρ] + Exc[ ˜ρ] (1.5.3)
where Ts is the single particle kinetic energy, VeN is the Coulomb
interac-tion between electrons and nuclei, VH is the Hartree potential, and Exc is
the exchange and correlation energy; a function that includes the electron correlation eects, and is de ned as:
Exc= T − Ts+ Vee− VH (1.5.4)
The expressions for the other quantities in eq. 1.5.2 are:
Ts[ ˜ρ] = − X i hφi| ∇2 2 |φii (1.5.5) VeN[ ˜ρ] = −X i,α hφi| Zα |~ri− ~Rα| |φii (1.5.6) VH[ ˜ρ] = X i<j hφiφj| 1 |~ri− ~rj||φiφji (1.5.7)
Actually, Eq. 1.5.3 is in formally exact, however the knowledge of the exact expression of the exchange and correlation functional is still unknown. The
development of DFT is based on nding accurate approximations of Exc[ρ].
The KS orbitals necessary to solve Eq. 1.5.1 are calculated solving the self-consistent KS equations:
ˆ
hKS(~r)φi(~r) = iφi(~r) (1.5.8)
Where ˆhKS is the KS operator, which has the following expression:
ˆ
hKS = −
∇2
2 + vN + vH + vxc (1.5.9)
vN is the nuclear external potential, vH is the classical electron-electron
re-pulsion, while vxc(~r) is the exchange and correlation potential and it comes
from the functional derivative of Exc[ρ] with respect to ρ:
The eects of electron correlation enter into the calculation through this po-tential. Dierent Density Functional Approximations(DFAs) lead to partially include dierent type of correlation. In the second chapter of this thesis, we will present an overview of the most commonly used DFAs.
1.6 Indicators of dynamic and nondynamic
cor-relation
Even if there is not a precise distinction between dynamic and nondynamic correlation, several attempts have been done in order to give an eective separation between dynamic and nondynamic correlation eects in quantum chemistry calculations. A clear distinction between dynamic and nondynamic correlation is not possible, but for some cases, it is possible to establish if a method includes mainly the eects of one or the other type of correlation. In the past, several tools have been developed in order to give a quantitative separation between these two contributions in quantum mechanic computa-tions.
For example, the T1 diagnostic of Lee et al. [Lee et al., 1989] gives a measure of the importance of the nondynamic correlation in coupled-cluster wave functions. Grimme and Hansen [Grimme and Hansen, 2015] de ned local indicators of electron correlation obtained by nite temperature DFT cal-culations. Others tried to separate the contribution of these two eects in the expression of the energies, as for example Cioslowski [Cioslowski, 1991], who suggested to calculate the FCI energy with a density-constrained ap-proach in which the FCI wavefunction is forced to reproduce the HF density,
EF CI[ρHF]. The dierence between EHF to EF CI[ρHF]contains the dynamics
correlation eects. On the contrary Valdarrena et al. [Valderrama et al., 1997] constrained an HF calculation with FCI density, considering the dierence
between EHF[ρF CI]and EHF a good estimation of non dynamic correlation.
Ecorr = EF CI − EHF = EDI + E I N D = E II N D+ E II D (1.6.1)
The idea of this decomposition is in Eq. 1.6.1, where the apex I stays for the Ciolowski decomposition, while the II stays for the Valderrana one. In gure 1.2 a schematic reproduction of these decompositions is shown. These methods are computational expensive, for this reason another proposal was given by Mok [Mok et al., 1996] which showed a separation of non dynamic correlation eects through the diererence of CASSCF method and HF.
Figure 1.2: Separation of dynamic and non dynamic correlation energy through the Ciolowski(I) and Valderrana(II) schemes.
The problem with the energy decomposition is the obtaining of accurate re-sults. Indeed, for both the components of electron correlation, they need expensive FCI calculations.
Ramos-Cordoba et al. [Ramos-Cordoba et al., 2016] proposed a partic-ular type of indicators to separate nondynamic and dynamic contributions. These indicators are constructed from the natural orbital occupancies. These indicators allow to express through a scalar quantity, the amount of dynamic and nondynamic correlation accounted for a given level of computation. The expressions of these indicators are the following:
IN D = 1 2 X i,σ nσi(1 − nσi) (1.6.3) ID = 1 4 X i,σ [nσi(1 − nσi)]1/2− 1 2 X i,σ nσi(1 − nσi) (1.6.4)
were IN D is the nondynamic indicator and ID is the dynamic one. These
indicators depend explicitly on the natural orbital occupancies, then they could be used for any method which allows the calculation of occupation numbers: any type of wavefunction methods, Density matrix functional the-ory or ensemble DFT with fractional occupancies. In terms of an orbital decomposition [Ramos-Cordoba and Matito, 2017] is possible to obtain also local descriptors. In fact, with such a decomposition it is possible to obtain real-space descriptors of electron correlation eects. There, the local weights
is introduced through the density of each natural orbital |φ(~r)|2 multiplied
by the corresponding indicators. These indicators are collected in Eqs. 1.6.5 and 1.6.6.
IN D(~r) =
1X
ID(~r) = 1 4 X i,σ [(nσi(1 − nσi))1/2− 2X i,σ nσi(1 − nσi)]|φσi(~r)|2 (1.6.6)
The integration of these quantities give back the global indicators 1.6.3 and 1.6.4. With these indicators, the region of the space in which dynamic and nondynamic correlation are more important are identi ed. This analysis could be eective for all the methods which show occupancies dierent from 0 and 1.
Chapter 2
Developments in DFT
The term Jacob's ladder was used by [Perdew and Schmidt, 2001] in order to organize the DFT Approximations (DFAs) on a scale of complexity and accuracy of the functionals. Every rung of the ladder represents a special type of DFA. Even if DFT has been very successful and it keeps growing there are some problems not yet solved in a comprehensive way.
In this chapter, a brief overview of the most commons DFAs used has been done.
2.1 Density functional approximations
The various proposal of exchange and correlation energy functionals are col-lected in Jacob's ladder, where every rung constitutes a dierent kind of approximation. Climbing the Jacob's ladder is equivalent to add more com-plexity to the expression of the exchange and correlation functional. Ex-change and correlation energy is viewed as a sum of a correlation and an exchange part, namely:
Exc= Ec+ Ex (2.1.1)
If Exc would be 0 the results of the calculation would be the analog of the
Hartree system, while, if Ecis null, the DFT is the analog of the HF method.
Hereafter, we review the rungs of the ladder, from the bottom to the top. LSDA At the bottom of the ladder, there is the local spin density ap-proximation (LSDA), which makes the assumption that the system behaves locally like a uniform electron gas. Therefore, the exchange term of Eq. 2.1.1
is expressed as the sum of all the local contributions, namely:
ExLSDA[ρ↑, ρ↓] =
Z
d~r(ρ↑(~r)HEGx,↑ (ρ↑(~r)) + ρ↓(~r)HEGx↓ (ρ↓(~r))) (2.1.2)
likewise the correlation will be:
EcLSDA[ρ↑, ρ↓] =
Z
d~rρ(~r)HEGc (ρ↑(~r), ρ↓(~r)) (2.1.3)
In the Eq. 2.1.2 there is the exchange energy per particle of an electron
gas with uniform spin densities ρ↑ and ρ↓. This quantity has been
calcu-lated analytically by Dirac [Dirac, 1930] and extended to atomic systems by Slater [Slater, 1951]. The correlation part of Eq. 2.1.3 is calculated by accurate numerical Monte Carlo simulations [Ceperley and Alder, 1980] and parameterized by Vosko et al. [Vosko et al., 1980]. This approximation works well for systems which have a slowly vary density, and for metals due to the properties of the uniform electron gas. Nevertheless, it has been demon-strated that this method fails in describing accurate HOMO and LUMO gap and overestimates bond energies.
Generalized Gradient approximation(GGA) and Meta-GGA In or-der to go beyond LDA approximation, it is necessary to add the eects of non-locality to the energy functional expression, considering the inhomo-geneity of the electronic density. This was done in the development of GGA functionals [Ziesche et al., 1998] in which exchange and correlation functional are obtained by an expansion of the density, where the 0th order is the LDA approximation, and the GGAs incorporate the rst term that depends on the gradient of the density.
ExcGGA[ρ] =
Z
d~rρ(~r)xc(ρ)LDAF (s) (2.1.4)
where the variable s is the reducent gradient variable, namely:
sα =
|∇ρα|
ρα
These modi cations generally improve LDA, obtaining better total energies, reaction barriers and partially correcting the understimation of bond lengths. This expansion is parameterized dierently for every functional developed, following dierent strategy: analytic derivation, physical considerations or tting from experimental data. However, the results are still inadequate for
most of the purposes. Further corrections based on higher derivatives lead to the meta-GGA's [Tao et al., 2003], which incorporate also the kinetic energy density: τ (~r) = 1 2 X i |∇φi(~r)|2 (2.1.5)
These kinds of functionals partially improve the results of GGA but they suer of the same problems.
Hybrid functionals-Adiabatic connection The next rung of Jacob's ladder is occupied by the hybrid functionals, whose purpose is to include a fraction of the nonlocal exchange as in the HF method [Becke, 1993b]. In the conventional hybrid description, the exchange and correlation energy is a combination of the exchange and correlation from a GGA and the exchange, for example:
Exchyb = ExcGGA+ a(ExHF − EGGA
x ) (2.1.6)
The justi cation of these methods is given by the adiabatic connection, which consists in the construction of a series of Hamiltonians depending on λ, a parameter multiplying the electron-electron interaction [Burke et al., 1997]. In this representation, the interaction of two electrons comes from an explicit
term which is λ/r12 and the corresponding external potential ˆVλ
s (~r) which
has to be adjusted for every value of λ in order to keep the density equal to the exact one, respecting the Hohenberg and Kohn theorem. The parameter
λ is called coupling constant, λ = 0 corresponding to a system without
electron-electron interaction and λ = 1 to the system with the real Coulomb interaction. The Hamiltonian of the system with the λ dependency could be expressed as: ˆ Hλ = −X i ∇2 i + X i ˆ Vsiλ(~r) +X i<j λ rij (2.1.7) The expectation value of the Hamiltonian will be:
Eλ = hψλ| ˆHλ|ψλi (2.1.8)
And because of the Hellmann Feynmann theorem, a de nition of the ex-change and correlation energy could be obtained:
Exc[ρ] = 1 2 Z 1 0 dλ Z d~rρ(~r)vxcλ = Z 1 0 dλWxcλ[ρ] (2.1.9) Where Wλ
xc[ρ] is the exchange and correlation energy for a system with the
The expression 2.1.9 could be used to obtain approximations to the ex-change and correlation energy. The development of these methods brought a boost in the application of DFT methods in computational chemistry, but they still fail for important properties such as nonlinear optical properties and for describing molecules at homolytic dissociation.
Range separated functionals These DFAs are based on the construction of the functionals separating the electron-electron interaction in two ranges of action, a short-range, and a long-range part [Henderson et al., 2008]. Most often this separation is done as:
1 r12 = 1 − erf (ω ~r12) r12 + erf (ω ~r12) r12 , (2.1.10)
where the rst term is the short-range part and the second is the long range one, and ω is the parameter which controls this separation. The main idea is to treat dierent ranges with dierent kind of functionals. These methods have been developed in order to correct the wrong asymptotic behavior of the exchange part of the previous functionals. Therefore, because the HF approximation gives a correct treatment of long-range exchange interaction for small molecules usually it is used for the developing of the long-range part of the functional
Recent developments In the last years, several developments with a sig-ni cant impact have been proposed. For example, highly parametrized func-tionals, which are characterized through the massive use of tting on exper-imental data, where the construction of the functionals loses any physical interpretation in favor of accurate results. As for example, Minnesota func-tionals [Zhao et al., 2005] and Head-Gordon works [Chai and Head-Gordon, 2008] who t the theoretical data.
Another rung of the Jacob's ladder currently under development involve un-occupied orbitals and eigenvalues, with the new kind of functionals such as double hybrids [Grimme, 2006] in which eects of perturbation theory are considered.
2.1.1 DFT problems and limitations
Some problems of DFT are actually still unsolved. In contrast with wavefunction methods DFT does not oer a systematic way of improving the Hamiltonian, the corrections of the functional are mostly speci c for the particular system or molecule, especially because many modern functionals are based on parameterization using speci c systems.
With respect to the HF theory DFT methods suer from the Self In-teraction. In the HF method, the exchange term in the energy cancels the interaction of an electron with itself. In DFT, where the exchange is approximate, Self Interaction could be not completely removed. Hy-brid methods partially reduce this de ciency, but it is still an open issue. DFAs fail in treating dispersion interactions in Van der Waals com-plexes and in describing H-bonds. Functionals with correction for dis-persions have been proposed, but are based on empirical parameters of such corrections. So far, no comprehensive density dependent solutions have been found.
Because of the single reference nature of the K-S wavefunction, in this framework, DFAs fail to describe systems with a multireference char-acter. Therefore, homolytic dissociation curves and diradical systems are poorly described with the existing KS-DFT approximations.
2.1.2 Electron correlation in DFT
The treatment of the interaction of electrons in DFT methods depends on the choice of the exchange and correlation functional. This functional is approximated in order to reproduce the real density, therefore it takes into account of eects of electron correlation in an unspeci ed way.
In particular, the exchange energy, which according to [Cremer, 2001] is the main source of the interaction energy, is not treated exactly by KS-DFT. In fact, dierently by the HF method, in which the explicit exchange is present, in DFT, it is approximated by the X term of the exchange and correlation functional. Instead, DFT methods, dierently from HF, take into account the Coulomb correlation. Dynamic correlation, which is due to the di -culty of treating electronic cusps, is mainly a short-range eect. This term is well approximated by DFT. On the other hand, nondynamic correlation,
which is mainly a long-range contribution, is a manifestation of the multi-con gurational character of the system itself. The KS framework, due to its intrinsic non-interacting form of the reference model, is not able to describe adequately these systems.
Cremer showed that the delocalization nature of hybrid functionals would lead to some better description of nondynamic correlation eects. Neverthe-less, due to a fortuitous cancelation of errors, in some cases, GGA functionals perform better.
2.2 DFT beyond KS approach
The intention to improve the results of KS-DFT of describing the systems with a multicon gurational character should involve the use of KS virtual or-bitals. They are introduced in the calculation by the removal of the constraint of integer occupation numbers. This modi cation leads to the construction of a non-idempotent 1-RDM, as should be for the real system.
Several ways have been tried in order to estimate the best occupation numbers and create a correct DFA for these systems. The use of the un-restricted framework could give fractional occupancies by breaking the spin symmetry. Indeed, projecting α and β spin orbitals in a natural orbital basis made possible to obtain the necessary fractional occupation numbers. This method generally brings to a lower energy, but it suers from spin contam-ination. Other approaches are based on a creation of multicon gurational DFT(MC-DFT), in which DFT orbitals are expanded in a multicon gura-tional structure, such as CAS-DFT [Ghosh et al., 2017] or GVB-DFT [Filatov et al., 2016].
Another interesting approach is the reconstruction of the occupation num-bers through certain parameters, for example, the work of Gruning [Gruning et al., 2003], who extracts the occupation number from a parameterized Fermi distribution.
In the next chapters, we will perform two computations in order to extract fractional occupation numbers. The rst comes from a multicon gurational GVB expansion with the coe cients tted from a QMC electronic density and the second from the broken symmetry DFT calculations.
Chapter 3
Fit of a Correlated density matrix
The DFT methods developed thus far give a good description of dynamic correlation, nevertheless they fail in the treatment of nondynamic one. The main reason for this failure is due to the single-determinant character of the wavefunction in the KS approach.
In this chapter it has been illustrated a way to go beyond the single-determinant formulation, generating a 1-RDM starting from the KS frame-work. Occupied and unoccupied KS orbitals are here used to construct a density from a GVB wavefunction. The parameters entering in the GVB form have been computed by tting an accurate correlated density derived from VMC calculation. The same parameters, thanks to the features of GVB density, give also a correlated N-representable 1-RDM. The simple shape of GVB wavefunction allows also to retrieve a pair density. We used this pair density to approximate the e-e interaction potential energy in order to esti-mate the energy value obtained from the Hohenberg-Kohn functional.
3.1 Reference density with QMC approach
The diagonal part of the 1-RDM is the electronic density of the system. In the following section, we will show how to obtain an accurate density from a VMC computation. In VMC, the Metropolis algorithm extracts the random con gurations by the importance sampling. For a given con guration, we
have a vector ~R = (~r1, ~r2, ..., ~rn)in a 3N dimensional space where ~ri indicates
the position of the i-th electron. These con gurations provide an approx-imation to the correlated electronic density. By assigning a Dirac's Delta function distribution for an electron at a given position and an in nite
num-ber of con gurations, we can write: ρV M C(~r) = lim M →∞ 1 M X M Nel X i=1 δ(~r − ~ri) (3.1.1)
In practical applications, there are a nite number of con gurations and in addition, we must approximate the Delta functions by a picked Gaussian distribution. Thus, for M con gurations, we write:
ρV M C(~r) ' 1 M M X k Nel X i=1 r λ3(~r) π3 e −λ(~ri)(~r−~ri)2 (3.1.2)
where the parameter λ has to be de ned in order to choose the correct width of the Gaussian distribution. Here, λ depends on the position in space where the Gaussian is centered. The higher the density, the higher the value of λ. In this way it is possible to create an electronic density which preserves the N-representability conditions:
Z
d~rρ(~r) = N ρ(~r) ≥ 0 (3.1.3)
3.1.1 Correlated 1-RDM
After the calculation of the electronic density that has been described above, we will estimate a suitable 1-RDM. The aim of this work is to go beyond a DFT calculation. The 1-RDM within a KS approach is idempotent (the occupation numbers are 0 or 1) so in order to compute a correlated density matrix, it is necessary to nd fractional occupation numbers of natural or-bitals. In terms of natural orbitals, the 1-RDM must have a diagonal form, which leads to
ρ(~r) =X
i
νi|φi(~r)|2 (3.1.4)
We attempted to t such a ρ(~r) on a pure methematical basis, but, even if we obtained reliable values for the occupation numbers, we reached unphysical values of the kinetic energy. Therefore, it is necessary to impose precise N-representability constraints on the 1-RDM above. For this reason, we moved to a GVB wavefunction.
GVB density matrix
We have chosen the GVB 1-RDM because the GVB wavefunction is mul-ticon gurational and the density matrix has a diagonal representation with
respect to the orbitals. Moreover, the GVB wavefunction uses Nel active orbitals, enough to capture the eects of nondynamic correlation. In or-der to overcome the single determinant character of the KS-DFT, the GVB wavefunction has been already computed in various contexts [Filatov et al., 2016]. In this work, the GVB wavefunction is described as a combination of Slater determinants, which, starting from the perfect pair(PP) reference adds all possible excitations of pairs of electrons, obtained by moving them from doubly occupied orbitals to the corresponding virtual ones. The number of
these functions would be N∗ = 2np where np is the number of pairs, and it is
equal to Nel/2in our case. If the number of electrons is assumed to be even,
we could express this wavefunction in a simpli ed form, like:
ΨGV B = N ˆA(b21+ λ1a21, b22+ λ2a22, ..., b2n+ λna2n) (3.1.5)
where the coe cients bn are the occupied Kohn-Sham localized bonding
or-bitals, meanwhile the an are the localized antibonding (virtual in KS
calcu-lation) orbitals. The operator ˆAis the usual antisymmetrizer operator. The
normalization N is xed by hΨGV B|ΨGV Bi = 1, thus we have np parameters,
namely the λi. The GVB 1-RDM becomes:
ρGV B(~r, ~r1) = N∗ X k Ck2X i∈k φi(~r)φ∗i(~r1) (3.1.6)
where the coe cients Ck depends on the level of excitation, and the φi are
bi or ai. The reference state is the PP one, and the corresponding coe cient
is:
C0 = N
the determinants with double excitations have a single parameter, namely
Ci = N λi
the quadruple excitations, and following all the others leads to:
Cij = N λiλj
Cij..n = N λiλj...λn
. From the equation 3.1.6, it is possible to recover the occupation number of the i orbital as:
νi = 2
X
k
where θik is a function which is 1 if the orbital i-th is one of the orbitals involved in the excitation k, otherwise 0, namely :
θik =
(
1 if i ∈ k
0 if i /∈ k (3.1.8)
In this work, DFT localized orbitals have been used as starting orbitals for the t of QMC electron density following equation 3.1.6. In particular, local-ized bonding orbitals are generated by a unitary transformation of occupied KS orbitals, while localized antibonding orbitals are generated by a unitary transformation of the unoccupied ones [Edmiston and Ruedenberg, 1965]. We have chosen the B3LYP functional for this purpose. We expect that the choice of the exchange and correlation functional would not be relevant for the resulting 1-RDM because the parameters are obtained through the QMC tting.
3.1.2 Pair density from the 1-RDM
The 2-RDM of the GVB wavefunction could be extracted knowing the
pa-rameters of the GVB 1-RDM by assuming that all λi are negative. This
Ansatz is in agreement with the fact that each antibonding orbital is used to correlate a given pair of electrons in each bond. For the t of the QMC electron density, we used the GVB pair density to compute T + Uee, namely the sum of kinetic energy and e-e repulsion potential energy. In fact, it would be useful to check if this quantity reaches a minimum for the optimized
pa-rameters λi. The expression of the Uee energy is:
Uee = N∗ X k Ck2 X i,j∈k (2 hφiφj| 1 r12 |φiφji−hφiφj| 1 r12 |φjφii)+ X k<l 0 CkCl X i∈k,j∈l hφiφj| 1 r12 |φiφji , (3.1.9)
where P0 means that the sum is limited to the con gurations that involves
a double substitution. This quantity is relevant in the de nition of the Ho-henberg and Kohn functional, namely:
F (ρ)HK = min
ψ→ρhψ|T + Uee|ψi (3.1.10)
3.2 The tting procedure
Having obtained the QMC density in the form of eq. 3.1.2, it is possible to proceed with the t. The t has been done with the least squares method,
minimizing the quantity ∆2 which is the square of the dierence between the
two densities integrated over the whole space. The expression of ∆2 is in the
following Eq. : ∆2min = min ρGV B [ Z d~r(ρQM C(~r) − ρGV B(~r))2] (3.2.1)
The integrand in Eq. 3.2.1 is explicitly written as:
|ρQM C(~r)|2 − 2 N∗ X k Ck X i∈k hρQM C(~r)|φ2i(~r)i + N∗ X k,l CkCl∗ X i,j∈k,l hφ2 j(~r)|φ2i(~r)i (3.2.2)
∆2 is minimized with respect to the parameters λn.
3.3 Test cases
We tested this procedure on a set of molecules that show evidence of nondy-namic correlation: para-benzyne, meta-benzyne and ortho-benzyne. These are biradicals isomers, therefore they show an open shell structure and an high reactivity.
Due to the presence of weakly paired electrons in the structure, the eects of nondynamic correlation are expected to be signi cant. The nondynamic char-acter arises due to the near-degeneracy of the HOMO and LUMO orbitals, which are localized at the position of the radicals. The distance between the unpaired electrons indicates the importance of nondynamic correlation in these molecules.
We already showed that in H2 dissociation nondynamic eects are
predomi-nant for electrons at large distances, likewise, in the diradicals, the distance of the weakly paired electrons in uences the presence of nondynamic correla-tion. In fact, as observed by Salem and Rowland [Salem and Rowland, 1972], if the radicals are far from each other, the molecular orbitals corresponding to the odds electrons will be localized. Therefore, there will be only a small overlap between these orbitals which leads to weak bonds. The resulting HOMO and LUMO orbital energies will be near-degenerate and the system will assume a multicon gurational character.
(a) p-benzyne (b) m-benzyne (c) o-benzyne
Figure 3.1: Geometries of the tested molecules
do not expect an evident multicon gurational character in this case.
3.4 Preparation of the VMC correlated density
The rst step is the VMC calculation. As trial wavefunction we choose a multicon gurational Slater-Jastrow wavefunction, which was subsequently optimized. The resulting optimized energy will be used as a reference for further calculations.
The initial setup The initial setup of the calculation has been provided by the software GAMESS [Schmidt et al., 1993]. We used a pseudopoten-tial [Burkatzki et al., 2007] and the corresponding VTZ basis set, in order to exclude the two inactive core electrons of the carbons. We determined an initial set of orbitals through an HF calculation and these orbitals have been reoptimized through a CAS(8,8) in order to improve the accuracy. The number of determinants in a CAS(8,8) is too high for a standard QMC cal-culation, so we have taken a threshold 0.01 on the coe cients to select the most important ones. The combination of the Slater determinants is than multiplicated by the Jastrow factor as in Eq.1.3.10 up to the fth order. The optimization of the wavefunction was made by the software CHAMP [Umri-gar and Filippi, 2018], and following the method of [Umri[Umri-gar et al., 2007]. Results of the calculation The results of the VMC calculations are col-lected in Table 3.1, where we show the ground state energy, the kinetic energy and the corresponding errors.
Molecule E0 δE T δT
o-benzyne -36.3440 0.0002 27.861 0.003
m-benzyne -36.3198 0.0002 27.585 0.003 p-benzyne -36.2872 0.0002 27.479 0.003
Table 3.1: Results of VMC calculations for ortho-, meta-, para-benzyne. The results are reported in Hartree: the ground state energy E0, its error δE, the kinetic energy T and its error δT .
3.4.1 Electronic density
For the GVB t of QMC density, we used a regular grid realized by the Gaussian code. According to Eq. 3.1.2 the value of the density on each part of the grid is:
ρ(~rg) = 1 M M X k Nel X i r w3(~r ik) π3 e −w(~rik)(~rg−~rik)2 (3.4.1)
The width w of each Gaussian has been chosen as depending on the cell of the grid. More precisely, w is proportional to the number of sampled points in the given cell as follows:
w(~rα) ∝
Nc(~rα)
d3
!23
(3.4.2)
where Nc(~rα) is the number of sampled con gurations, and 2d is the side of
the cube. The density of the system has to be greater than 0
ρ(~r) ≥ 0 (3.4.3)
And satisfy the normalization condition: Z
d~rρ(~r) = N (3.4.4)
The densities obtained through the interpolation for the three molecules un-der investigation are shown in Fig. 3.2 .
3.5 Fitting of the density
The GVB wavefunction was prepared by a combination of single particle orbitals generated by a B3LYP computation, this calculation has been
per-(a) p-benzyne (b) m-benzyne (c) o-benzyne
Figure 3.2: QMC electronic densities for the dierent molecules used. The orbitals used for the computation of the 1-RDM were 28, namely 14 bonding and 14 antibonding orbitals, in order to reproduce the 28 valence electrons of benzyne molecules. These orbitals have been collected from the
output of Gaussian and then used for the creation of the function ∆2.
The minimization of ∆2 was performed in two steps: in the rst step we
used a stochastic procedure and in the second a steepest descendent method. The rst step is a Metropolis sampling, in which in every iteration the set
of variational parameters pi are modi ed randomly with a value in the range
[−δ
2 ,
δ
2], as follows:
pi = p0i+ δ(0.5 − zr)
where zr is a random number between 0 and 1. Given the step, the sampling
is governed by the Boltzmann distribution:
PB = N e−β∆
2
(3.5.1) The goal is to explore as much as possible the space of parameters, in order to prevent the algorithm from getting stuck with a local minimum. For this reason, it is necessary to properly tune the quantities δ and β. An estimator of the e ciency of the sampling for the Metropolis process is the acceptance (A), which is de ned as the ratio between the number of accepted moves and the total ones.
A = nacc
Ntotal
(3.5.2) The Metropolis algorithm is considered e cient if the acceptance is between 0.4 and 0.6. The parameter β needs to be tuned with the number of cycles
Ntotal in order to stay the acceptance in this interval. In Table 3.2 we show
the global parameters used in the thesis.
In the second step, we proceded with a steepest descendent method, starting
from the point in the parameter space where ∆2 is minimum after the rst
δ β Ntotal δs 0.025 5000 40000 0.001
Table 3.2: Parameters chosen for the minimization of p-, m-, o-benzyne.
10000 20000 30000 40000 50000 60000 70000 Number of steps 0.0072 0.0074 0.0076 0.0078 0.0080 0.0082 0.0084 0.0086 0.0088 2
p-benzyne
Figure 3.3: Variation of ∆2 along the optimization steps for p-benzyne.
δ (see Table 3.2) in order to nd the global minimum. In thus steepest
de-scendent, we allow the mixing of the pair bonding and antibonding orbitals. With the set of parameters chosen, we obtained an acceptance of 0.54 for p-benzyne, 0.53 for m-benzyne and 0.50 for o-benzyne. In order to be con-sistent, the initial parameters have been set to 0, corresponding to the KS approach.
In every step of the calculation, the T and Uee were evaluated. Particularly
interesting is the kinetic energy because it could be compared with the
ac-curate VMC one. The variations of ∆2 along the optimization are shown in
Figs. 3.3, 3.4 and 3.5.
The nal results of the minimization are shown in Table 3.3. As it is possible to observe comparing the results of Table 3.3 with the ones obtained in Table 3.1, the nal value of the kinetic energy is understimated for all the systems of 0.5 Hartree. In Table 3.4 the values of the occupation num-bers obtained from the calculation are shown. The dierences from the KS reference can be observed just for the orbitals which de ne the diradicals, which are fractionals and between 0 and 2. This dierence is more evident
10000 20000 30000 40000 50000 60000 70000 Number of steps 0.0105 0.0106 0.0107 0.0108 0.0109 0.0110 0.0111 0.0112 2
m-benzyne
Figure 3.4: Variation of ∆2 along the optimization steps for m-benzyne.
10000 20000 30000 40000 50000 60000 70000 Number of steps 0.0098 0.0100 0.0102 0.0104 0.0106 2
o-benzyne
Molecule ∆2 T + ˜˜ Uee T˜
o-benzyne 0.0098 135.43 27.37
m-benzyne 0.0105 134.59 27.10
p-benzyne 0.0072 133.96 27.12
Table 3.3: Results of the minimization for p-,m-,o- benzyne. Final values for
the ∆2, the function ˜U
ee+ ˜T and the kinetic energy ˜T.
10000 20000 30000 40000 50000 60000 70000 Number of steps 133.95 134.00 134.05 134.10 134.15 134.20 134.25 134.30 134.35
T+U functional [Hartree]
p-benzyne
Figure 3.6: Behavior of the T + Uee functional along the optimization for
p-benzyne.
for systems which display the largest amount of correlation.
In Figs.3.6, 3.7 and 3.8 it is shown how the function T + Uee is modi ed
along with the minimization of ∆2. In particular, it is possible to observe
that this value decreases with the improvement of the tting, but the nal value is not exactly the minimum.
3.6 Indicators of nondynamic correlation
The occupation numbers obtained show fractional occupancies for the or-bitals which are responsible for the multicon gurational character, therefore, we moved from the KS electron density to a correlated one.
N° p-benzyne m-benzyne o-benzyne 1 2.00 2.00 2.00 2 2.00 2.00 2.00 3 2.00 2.00 2.00 4 2.00 2.00 2.00 5 2.00 2.00 2.00 6 2.00 2.00 2.00 7 2.00 2.00 2.00 8 2.00 2.00 2.00 9 2.00 2.00 2.00 10 2.00 2.00 2.00 11 2.00 2.00 1.94 12 2.00 2.00 2.00 13 1.99 2.00 2.00 14 0.86 1.83 2.00 15 1.14 0.17 0.00 16 0.00 0.00 0.00 17 0.01 0.00 0.06 18 0.00 0.00 0.00 19 0.00 0.00 0.00 20 0.00 0.00 0.00 21 0.00 0.00 0.00 22 0.00 0.00 0.00 23 0.00 0.00 0.00 24 0.00 0.00 0.00 25 0.00 0.00 0.00 26 0.00 0.00 0.00 27 0.00 0.00 0.00 28 0.00 0.00 0.00
10000 20000 30000 40000 50000 60000 70000 Number of steps 134.58 134.60 134.62 134.64 134.66
T+U functional [Hartree]
m-benzyne
Figure 3.7: Behavior of the T + Uee functional along the optimization for
p-benzyne. 10000 20000 30000 40000 50000 60000 70000 Number of steps 135.42 135.44 135.46 135.48 135.50 135.52
T+U functional [Hartree]
o-benzyne
Figure 3.8: Behavior of the T + Uee functional along the optimization for
