The Hartree-Fock reference function

We develop the perturbation theory by separating the N-electron Hamiltonian (1.1.2) into two parts  

It is possible to obtain a virtually exact ground-state wavefunction for this model, using the methods of Chapter 6. The resultant function d g is taken as the leading term or reference function in the Cl expansion 

The existence of a convenient reference state means that when using second-quantization methods we do not need to start always from the vacuum state vac) we can instead use 

To this end we shall adopt systematically the usual conventions labels i, j, k, I will refer to spin-orbitals occupied in labels m, n, p, q to those unoccupied in (i-e. virtual) and r, s, t, u to those of either type. Briefly, r/ , and ip are respectively occupied and virtual (or excited ) spin-orbitals. We note that a and a, are both, in a sense, creation operators relative to the reference state ai, creates a particle in the excited spin-orbital while a creates a hole in the originally occupied spin-orbital Similarly a and aj are both annihilation operators in the sense that a destroys a particle in the excited spin-orbital ipm while aJ destroys a hole by refilling the spin-orbital i/ , of pQ. The reference state represented by 0) is thus a new vacuum state , which contains neither holes nor particles. This point of view may be emphasized by a change of notation that is sometimes used  

These new operators describe the creation and annihilation of quasiparticles (i.e. holes below a certain energy threshold, particles above), and their anticommutation properties are easily seen to be the same as for the original operators  

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Unlike the second-order and third-order energy diagrams, the fourth-order diagrams can involve intermediate states which are singly-excited, doubly-excited, triply-excited, and quadruply-excited with respect to the Hartree-Fock reference function.8 130... 

Finally, there are seven fourth-order terms which involve an intermediate state which is quadruply-excited with respect to the Hartree-Fock reference function. These diagrams are shown in Figure 8. Time reversal relates diagrams (Bq, Cq), (Dq, Gq), and (Eq, Fq). Explicitly, diagrams Aq, Bq, and Cq, for example, correspond to the expressions... 

H(0 )n is the normal ordered form of Hamiltonian of the solute in the presence of the apparent charges determined by the Hartree-Fock reference function... 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

The many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. 

By reference to the expressions of dy -d>r, in Exercise 4.2, the expression of the Hartree—Fock wave function in terms of VB structures is... 

In this section, we propose to illustrate how the availability of the CRAY has assisted progress in the area of molecular electronic structure. We shall concentrate on two recent advances, namely, the evaluation of the components of the correlation energy which may be associated with higher order excitations, in particular triple-excitations with respect to a single-determinantal, Hartree-Fock reference function, and the construction of the large basis sets which are ultimately going to be necessary to perform calculations of chemical accuracy, that is one millihartree. 

Figure 11 Order of perturbation at which various blocks of the configuration mixing matrix first contribute to the energy (a) for Hartree-Fock reference function (b) for ibare-nucleus> reference function (c) for reference function constructed from Brueckner orbitals... 

There are a number of ways in which one may begin to correct the Hartree-Fock wave function so as to include electron correlation ". The simplest in concept is configuration interaction (Cl) which takes the Hartree-Fock solution as a starting point, or reference con-... 

Tel. 904-392-1597, fax. 904-392-8722, e-mail aces2 qtp.ufl.edu Ab initio molecular orbital code specializing in the evaluation of the correlation energy using many-body perturbation theory and coupled-cluster theory. Analytic gradients of the energy available at MBPT(2), MBPT(3), MBPT(4), and CC levels for restricted and unrestricted Hartree-Fock reference functions. MBPT(2) and CC gradients. Also available for ROHE reference functions. UNIX workstations. 

D. Jayatilaka and T. J. Lee, Chem. Phys. Lett., 199, 211 (1992). The Form of Spin Orbitals for Open-Shell Restricted Hartree-Fock Reference Functions. 

The classification of CSFs into single, double,... excitations is straightforward and unambiguous for a closed-shell Hartree-Fock reference function. In open-shell or multireference cases, there are more possibilities for defining these excited CSFs, some of which interact with the reference space in the lowest order of perturbation theory, and some of which do not. It is very common to exclude the latter excitations. This is termed restricting the wave function to the first-order interacting space. ... 

Skipping the reference to the electronic (e) subscripts throughout Eqs. (4.250)-(4.262), the Hartree-Fock trial functional can firstly be arranged as by the optimization procedure (Putz Chiriac, 2008)... 

Though used in some semiempirical applications by Paldus and Cizek [11] and one ab initio study [12] (see later), the CCD equations were not implemented into general purpose programs until 1978 by me and Purvis [5] and Pople et al. [13]. This general implementation included allowing for the open-shell case subject to an unrestricted Hartree-Fock reference function. 

We will begin this chapter by constructing determinantal trial functions from the Hartree-Fock molecular orbitals, obtained by solving Roothaan s equations. It will prove convenient to describe the possible N-electron functions by specifying how they differ from the Hartree-Fock wave function Fo Wave functions that differ from Tq by w spin orbitals are called n-tuply excited determinants. We then consider the structure of the full Cl matrix, which is simply the Hamiltonian matrix in the basis of all possible N-electron functions formed by replacing none, one, two,... all the way up to N spin orbitals in Section 4.2 we consider various approximations to the full Cl matrix obtained by truncating the many-electron trial function at some excitation level. In particular, we discuss, in some detail, a form of truncated Cl in which the trial function contains determinants which differ from To by at most two spin orbitals. Such a calculation is referred to as singly and doubly excited Cl (SDCI). 

In the second quantization formalism we introduce a reference state for the system under study, which is a Slater determinant (usually the Hartree-Fock wave function) composed of N orthonormal spinorbitals, where N is the number of electrons. This function will be denoted in short by o or in a more detailed way by notation means that we have to do with a normal-... 

Several recent publications have proven [54-56] the validity of equation (5) in connection with the Harris functional, which is often used in electronic structure calculations of molecules, surfaces and other condensed-matter systems. We note that, in the context of nuclear physics, Strutinsky had earlier proven [38] the validity of Eq. (5), with the difference that he utilized the Hartree-Fock (HF) functional instead of the KS-LDA one. In the nuclear physics literature, the HF version of Eq. (5) is referred to as the Strutinsky theorem . 

At this point the reader may be wondering where it all ends. In theory, the answer is never. To construct a complete basis set, capable of exactly representing the Hartree-Fock wave function for any molecule, it would be necessary to include an infinite number of functions of each symmetry type (s,p,d,f,. ..). This is sometimes referred to as the Hartree-Fock limit. For an in-depth examination of this issue the reader is referred to representative work by McDowelP and Klahn. Although a rigorous examination of completeness is beyond the scope of the present treatment, it is helpful to consider a more practical definition of completeness that allows for real world limitations. We thus arrive at the notion of effectively complete basis sets. 

Approximate many-electron wave functions are then constructed from the Hartree-Fock reference and the excited-state configurations via some sort of expansion (e.g., a linear expansion in Cl theory, an exponential expansion in CC theory, or a perturbative power series expansion in MBPT). When all possible excitations have been incorporated (S, D, T,. .., for an -electron system), one obtains the exact solution to the nonrelativistic electronic Schrodinger equation for a given AO basis set. This -particle limit is typically referred to as the full Cl (FCI) limit (which is equivalent to the full CC limit). As Figure 5 illustrates, several WFT methods can, at least in principle, converge to the FCI limit by systematically increasing the excitation level (or perturbation order) included in the expansion technique. 

The Hartree-Fock reference state HF) is obtained by the stationarity condition of Hartree-Fock free energy functional Ohf (Cammi and Tomasi 1995a)... 

The use of a Hartree-Fock reference function is ubiquitous in molecular electronic structure theory because of the beneficial computational consequences of the orthogonality of the Hartree-Fock molecular orbitals. However, many quantum chemical studies require the use of a multi-reference formalism. For example, studies of systems involving bond breaking processes almost invariably require the use of a reference function constructed as a linear combination of a number of reference functions. For cases where electron correlation effects are large and, in particular, when the Hartree-Fock model gives qualitatively incorrect results, the system is said to be strongly correlated. 

The cluster operator f of the truncated wave function contains only a subset of the full set of excitation operators. For example, in a CCSD calculation, we include in the cluster operator only the single and double excitations out of the Hartree-Fock reference state. The projection manifold Ipi) is initially chosen in the usual manner - that is, such that it spans all the states that may he created when the cluster operator works on the RHF reference state. The total number of amplitudes (and states in the projection manifold) is denoted hy A d-... 

Obtain the explicit form of the second-order correction 1 2) to the wavefunction for a system with Hartree-Fock reference function. [Hint Proceed as in the derivation of (9.4.18).]... 

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