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Accurate electron correlation: hints on post-HF methods
A precise quantication of the electron correlation can be exploited by using post- HF methods. These are, together with Quantum Monte Carlo (QMC), the most accurate computational techniques available nowadays, because they are able to include electron correlation by considering excited-states congurations within the solid framework of wave function theory. In this section, only basic concepts for the most popular post-HF techniques are given, since they were not mainly used to carry out calculations in the present work of thesis because of the large size of the investigated systems. However, we were able to run local-MP2 calculations assuming them as a benchmarking reference for the study of the relative stability of oxalyl dihydrazide (see Chapter 6.2). Since a nite basis-set expansion is used, the HF method yields to a nite set of M spatial wave functions { this constitutes the HF basis-set truncation error. Moreover, an innite number of detors can (in theory) be used to describe the N occupied SOs of the ground-state wave function; the HF method, instead, uses only one determinant. We also recall that 2M N virtual SOs can be dened from M spatial wave functions.
Hence, multiple detors can describe both occupied and virtual SOs. These last can be classied taking as reference the single determinantal HF wave function, only by highlighting the promotion of electrons from ground state to excited congurations. Dening the ground state HF detor as j 0i, one can express a singly-excited determinant as j r ai, where one electron is promoted from the occupied orbital a to the virtual orbital r. A doubly-excited determinant can be similarly dened as j rs abi, whereas detors for higher order excitations can be derived in the same way { these are called conguration state functions (CSFs). When the expressions for the wave function and the Hamiltonian operator are dened, every post-HF method can be solved self-consistently by applying the Variational Principle as in the case of HF.
Mller-Plesset Perturbation Theory
The Mller-Plesset (MP) perturbation theory is a particular case of the Rayleigh- Schrodinger (RS) perturbation theory. The latter denes the total electronic Hamiltonian as the sum of an unperturbed Hamiltonian operator bH0 and a small perturbation bH 0: bH= bH 0 + bH 0 (1.3.9) where is a small, arbitrary real parameter that weights the perturbation. This last corresponds to the electron correlation potential. regulates the order of perturbation, entering in the power series that express the form of the energy and the wave function. If we consider also a second order perturbation term, we can write: bH = bH0 + bH 0 + 2bH 00.
Density operators, reduced and spinless density matrices
In general, the description of a purey quantum state can be given by the following denition of probability: N(x1x2 xN) N(x1x2 xN) (2.1.1).
which is associated with the solutions of the Schrodinger equation for an N-electron system. The set x1x2 xN includes all the space and spin coordinates for each electron. We dene a density operator as: b N = j Nih Nj (2.1.2) that { for two distinct sets of coordinates fxig, fx0 ig { acts in the following way: hx0 1×0 2 x0 Njb Njx1x2 xNi = (x0 1×0 2 x0 Nj Nih Njx1x2 xN) = = N(x0 1×0 2 x0 N) N(x1x2 xN) (2.1.3).
Thus, it corresponds to a projection operator. For a normalized set of N, we have: tr(b N) = Z N(xN) N(xN)dxN = 1 (2.1.4). this means that, in the matrix form, the trace (sum of diagonal elements) is normalized to the total number of electrons in the system.
The Hohenberg-Kohn theorems
The Hohenberg-Kohn (HK) theorems [1] (1964) recall that { for any N-electron system { the external potential v(r) of the Hamiltonian (equation 1.1.3) denes the whole nuclear frame, and together with the number of electrons N, it determines all the ground-state properties. Instead of using N and v(r), the rst HK theorem assumes the electron density (r) to be used as the fundamental variable.
v- and N-representability of the electron density
Being the electron density minimized, it must satisfy two conditions, namely the v-representability and N-representability. The rst one requests that such density truly corresponds to the ground-state density of a potential v(r) { which is not necessarily a Coulomb potential { associated with the antisymmetric ground-state wave function of the Hamiltonian dened in eq. 1.1.2. The rst HK theorem can be stated, in other words, as the fact that there is a unique mapping between a v-representable density and the ground-state wave function.
This leads to the determination of all the other ground-state properties. The universal functional { FHK { in particular has to be v-representable. Since it enters in the denition of the variational principle, it is clear that both the HK theorems, so the DFT itself, request a trial density to be v-representable. Unfortunately, the conditions for a density to be v-representable are still unknown. On the other hand, some reasonable expressions for trial densities have been demonstrated to be non v-representable.
However, DFT can be formulated in a way { vide infra, Sec. 2.3 { that the density only requires to satisfy the N-representability condition, i.e. to be obtained from an antisymmetric wave function. This constitutes a weaker condition with respect to the v-representability, because it is a necessary condition for the latter. Mathematically, the properties to be satised are: (r) 0 ; Z (r)d(r) = N ; Z jr(r)1=2j2d(r) < 1 (2.2.9).
The Levy-Lieb constrained-search
We have established the univocal correspondence between the ground state electron density and the ground-state wave function but still not the vice versa. In fact, the problem is that an innite number of antisymmetric wave functions { not only from the ground-state { can yield to the same density. If one chooses a proper trial wave function that integrates to the ground-state 0(r), e.g. j~ 0i, he does have to distinguish it from the true ground-state j 0i.
This can be achieved by the procedure of Levy [3, 4], which makes use of the minimum energy principle for the ground-state: h~ 0jbH j~ 0i h 0jbH j 0i = E0.
The adiabatic connection and the XC hole
The denition of the exchange-correlation hole (XC hole), through the adiabatic connection method, consists of a powerful approach to link a noninteracting system to the interacting one in the framework of KS-DFT { and a rst step to build approximations for the exchange-correlation functional.
The universal functional dened in Sec. 2.2 for the constrained search of the electron density (r) can be rewritten as: F[(r)] = min ! h j b T + bVeej i = h j b T + bVeej i (2.5.1). where is a parameter that controls the strength of the interelectronic interactions { the coupling strength parameter { and represents the wave function which minimizes h b T + bVeei. Given that connects the noninteracting system ( = 0) to the fully interacting one ( = 1), we may express the corresponding two forms of F[] as: F1[(r)] = T[(r)] + Vee[(r)].
The Self-Interaction error and the HF/KS method
Even though standard KS-DFT contains an exchange term, it is aected by the self-interaction error, SIE (which is not present in HF, as discussed in 1.1.6). It is caused by the local form of the exchange interaction, which should be nonlocal, as the HF exact exchange is.
The SIE disappears only in one-electron systems, where the self-Coulomb repulsion is exactly canceled by the self-exchange term. This corresponds to an intrinsic self- interaction correction (SIC).
In many-electron systems, the one-electron SIE leads to an articial stabilization of delocalized states, whereas the many-electron SIE (N-SIE { that is due to the nonlinear dependence of energy with respect of the particle number, for fractionally charged systems) leads to an incorrect description of band gaps, ionization potentials, electron anities and other important properties. The SIE issue could be partially solved by substituting the local KS-DFT exchange with a percentage of { or the full { HF nonlocal exchange. This gives rise to the hybrid HF/KS method (see also Sec. 3.3): the standard HF scheme can be thought as a DFT method without correlation, with an exact, nonlocal description of the exchange potential. Thus, a full inclusion of the latter in DFT leads to a more rigorous physics { and only the correlation potential needs to be approximated. However, notice that since the KS-DFT requires the exchange potential to be local, the nonlocal HF one is not the exact exchange contribution within the framework of DFT. This means that the KS and HF/KS exchange must be related in some way (see e.g. [6]).
Exchange-correlation functional approximations
DFT is formally exact, but in practice approximations are needed to dene the unknown exchange-correlation functional { Exc[(r)]. Many approximated expressions have been proposed in the past { and still now they are the main matter of development in DFT.
Exc[(r)] approximations can be classied hierarchically following the so-called Jacob’s Ladder, proposed by Perdew [7] (Fig. 2.1), from the oldest and simplest ones to more recent and complex ones.
Table of contents :
Notice
Introduction
Theoretical background
Part I – Quantum theories for ground-state chemistry
1 Wave function theory
1.1 The Hartree-Fock Method
1.1.1 The Hartree-Fock equations
1.1.2 How to derive the HF equations
1.1.3 Spinless HF equations
1.1.4 Coulomb operator
1.1.5 Fermi or Exchange operator
1.1.6 The Self-Interaction (SI) Error and Why the HF scheme is SI-free
1.1.7 RPN linearization of the HF equations
1.1.8 Switching to the standard form
1.1.9 Orbital energies
1.1.10 Koopman’s theorem
1.2 The correlation energy problem
1.3 Accurate electron correlation: hints on post-HF methods
1.3.1 Conguration Interaction
1.3.2 Coupled Cluster
1.3.3 Mller-Plesset Perturbation Theory
1.3.4 Multireference Ansatze
2 Density Functional Theory
2.1 Density operators, reduced and spinless density matrices
2.2 The Hohenberg-Kohn theorems
2.2.1 v- and N-representability of the electron density
2.3 The Levy-Lieb constrained-search
2.4 The Kohn-Sham formulation of DFT
2.5 The adiabatic connection and the XC hole
2.6 The Self-Interaction error and the HF/KS method
2.7 Exchange-correlation functional approximations
2.8 Global hybrid functionals
2.9 Including electron correlation into DFT
2.9.1 The Stairway to Heaven of dispersion
3 Periodic quantum calculations
3.1 Reciprocal lattice and rst Brillouin Zone
3.2 From local GTFs Basis-Set to Bloch Functions
3.3 Crystalline Orbitals
Part II – Excited-state properties and multiscale methods
4 Time-Dependent Density Functional Theory
4.1 Time-dependent KS equations: RG theorems
4.2 Linear response theory in TD-DFT
4.3 Excitation energies and oscillator strengths
5 Multiscale methods
5.1 Environmental eects
Applications
Conspectus
References
6 Polymorphism and stability of p-diiodobenzene and oxalyl dihy- drazide
6.1 p-diiodobenzene (p-DIB)
6.1.1 Relative stability of p-DIB
6.1.2 Structural parameters
6.2 Oxalyl dihydrazide (ODH)
6.2.1 Crystal structure optimizations
6.2.2 Relative Stability of ODH
7 Unveiling the polymorphism of [(p-cymene)Ru(kN-INA)Cl2]
7.1 Introduction
7.2 Computational Methodologies
7.3 1 and 1H2O crystal forms
7.3.1 Structural parameters: cell constants.
7.3.2 Intra- and inter- molecular parameters.
7.3.3 13C and 1H NMR calculations.
7.4 1 polymorph
7.4.1 Structural parameters.
7.4.2 13C and 1H NMR calculations.
7.5 1 polymorph
7.5.1 Comparison with the other forms
7.5.2 Structural parameters
7.5.3 13C and 1H NMR calculations
8 Investigating the optical features of thermochromic salicylidene anyline
8.1 Introduction
8.2 Computational protocol
8.3 polymorph: structural and electronic behaviour
8.3.1 Intra- and inter- molecular properties
8.4 Optical features: UV-Visible absorption
References
