Slater determinant excited

Figure 4.1 Excited Slater determinants generated from a HF reference...

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... 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

The T, operator aeting on a HF reference wave function generates all tth excited Slater determinants. 

Cl methods [21] add a certain number of excited Slater determinants, usually selected by the excitation type (e.g. single, double, triple excitations), which were initially not present in the CASSCF wave function, and treat them in a non-perturbative way. Inclusion of additional configurations allows for more degrees of freedom in the total wave function, thus improving its overall description. These methods are extremely costly and therefore, are only applicable to small systems. Among this class of methods, DDCI (difference-dedicated configuration interaction) [22] and CISD (single- and double excitations) [21] are the most popular. 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

Exchange operator, 61 Excited Slater determinant, 99 Excited states, 147------------... 

Here, the Ti operator when working on Pq affords the ith excited Slater determinants. In practical, CC calculations T of (25) is truncated. Thus keeping T + T2 gives rise to CCSD whereas the addition of 7) and subsequently T4 leads to CCSDT and CCSDTQ, respectively. The CCSD scheme which scales as (nef is used routinely for up to 100 electrons. It is considered as the most accurate method for metal complexes in those cases where the reference HF determinant Po affords a reasonable description. CCSDT and CCSDTQ scales as (nef and (ne)l(>, respectively, they can only be used for very small systems. 

Here (pj(k, r) is the kth CO of the band j. Finally in equation (26) is a doubly excited Slater determinant in which was excited from the filled Wannier functions w,/f, wjjt, to the originally unfilled ones w, Wbj, . 

Projection of equation (29) on the space of doubly excited Slater determinants (expressed by Wannier functions) gives... 

The excited Slater determinants are generated by removing electrons from occupied orbitals, and placing them in virtual orbitals. The number of excited SDs is thus a combinatorial problem, and therefore increases factorially with the number of electrons and basis functions. Consider for example a system such as H2O with a 6-31G(d) basis. For the purpose of illustration, let us for a moment return to the spin-orbital description. There are 10 electrons and 38 spin-MOs, of which 10 are occupied and 28 are empty. There are possible ways of selecting n electrons out of the 10 occupied orbitals, and ways of distributing them in the 28 empty orbitals. The number of excited states for a given excitation level is thus Kw -K2s, and the total number of excited determinants will be a sum over 10 such terms. This is also equivalent to. Ksg io, the total number of ways 10 electrons can be distributed in 38 orbitals. 

We have here neglected the normalization constants for both the MOs and the deter-minantal wave function. The bar above the MO indicates that the electron has a spin function, no bar indicates an a spin function. In this basis, there are one doubly (i) and four singly excited Slater determinants (2 5). 

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