Variation Principle Linear Expansion

The variation principle is used to get the best possible approximation of a function. In almost all applications of quantum chemistry, the coefficients are optimized in a linear expansion. In Chapter 2, we will use the variation principle to derive the Hartree-Fock equation. 

Assume that we know the eigenfunctions T i and eigenvalues of the Hamiltonian H. We also assume that an arbitrary function O can be expanded in terms of these eigenfunctions (in principle)  

The set TJ is assumed to be orthonormal. The overlapping case is treated in Appendix 5. We require from d that it is normalized  

We will only show that the energy difference 8E between the Hamiltonian expectation valne for O and the lowest energy eigenvalue of H is always positive. The difference may be written as 

This proves the theorem and defines the term best possible. In optimization, we can never end up with an energy expectation value smaller than the lowest exact eigenvalue of H. The expectation value is always an upper bound to the true eigenvalue and this is the first part of the variation principle. 

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Whereas the singles occur to fourth order, the doubles appear only quadratically in these exjxessions. Moreover, from (13.2.39), it is easily verified that, except for the singles and the doubles, the amplitudes of the highest excitation level always occur oidy linearly in the coupled-cluster equations. These addifional simplifications in the coupled-cluster equations - beyond what is dictated by the termination of the BCH expansion (13.2.37) - occur because of the restrictions on the excitation levels in the projection space (/r,j. If instead we had calculated the energy fhrm the variation principle, the expansion of the exponentials would not terminate (except, of coinse, for the fact that we have only a finite number of electrons to excite from the occupied spin orbitals). The evaluation... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

From the perspective of variational principles, the idea embodied in the finite element approach is similar to that in other schemes identify that particular linear combination of basis functions that is best in a sense to be defined below. The approximate solution of interest is built up as linear combinations of basis functions, and the crucial question becomes how to formulate the original boundary value problem in terms of the relevant expansion coefficients. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

We have thus been able to construct a wave function that describes the qualitative behavior of the electronic stmcture for all internuclear distances. The price we have paid is to leave the single configurational description and construct the wave function as a linear combination of several configurations (determinants) with expansion coefficients to be determined by the variational principle together with the molecular orbital coefficients. This is the multiconfigurational approach in quantum chemistry. Before we end this section let us take a look at a more complex chemical bond, that in the Cr2 molecule. 

The subscript / has been omitted from A fc(r) for notational simplicity. The action S becomes a quadratic function of the expansion coefficients (p<, Qi) when the expansions (118) are substituted into the action integral. The variational principle then leads to a system of linear equations for the expansion coefficients,... 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

The Coulomb repulsion between the two electrons is l/r,2, where r,2 is the distance between the electrons. We are going to use the variation principle on a linear expansion. All basis functions contain one (or more) exponential function where the effective nuclear charge is obtained from screening theory (Section 2.2.6). Since there is one other electron in the same shell, the screening factor is equal to S = 0.35. A rather good function to approximate the Is orbital is thus exp[-(2 - 0.35)r] = exp(-1.65r). 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

The studies show that the observed crystal volume is in fact composed of the fractional contributions from the unit cell volumes of the HS and LS isomers of the compound and a linear volume change with temperature as expressed in Eq. (128). Similarly, the observed lattice constants are formed from a deformation contribution proportional to the HS fraction and a contribution from thermal expansion following Eq. (131). This is a convincing demonstration that it is the internal variation of the molecular units occurring in the course of the spin-state transition which determines, at least in principle, the observed crystal properties. 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

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