HF determinant

It is particularly desirable to use MCSCF or MRCI if the HF wave function yield a poor qualitative description of the system. This can be determined by examining the weight of the HF reference determinant in a single-reference Cl calculation. If the HF determinant weight is less than about 0.9, then it is a poor description of the system, indicating the need for either a multiple-reference calculation or triple and quadruple excitations in a single-reference calculation. 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

The number of excited determinants thus grows factorially with the size of the basis set. Many of these excited determinants will of course have different spin multiplicity (triplet, quintet etc. states for a singlet HF determinant), and can therefore be left out in the calculation. Generating only the singlet CSFs, the number of configurations at each excitation level is shown in Table 4.1. 

Another way of viewing spin contamination is to write the UHF wave function as a linear combination of pure R(0)HF determinants, e.g. for a singlet state. 

Since the UHF wave function is multi-determinantal in terms of R(0)HF determinants, it follows that it to some extent includes electron correlation (relative to the RHF reference). 

The zero-order wave function is the HF determinant, and the zero-order energy is just a sum of MO energies. The first-order energy correction is the average of the perturbation operator over the zero-order wave function (eq. (4.36)). 

In order to achieve a high aceuraey, it would seem desirable to explicitly include terms in the wave functions which are linear in the intereleetronie distanee. This is the idea in the R12 methods developed by Kutzelnigg and co-workers. The first order correction to the HF wave funetion only involves doubly exeited determinants (eqs. (4.35) and (4.37)). In R12 methods additional terms are included which essentially are the HF determinant multiplied with faetors. 

In conclusion, we observe that many writers in the modern literature seem to agree about the convenience of the definition (Eq. 11.67), but that there has also been a great deal of confusion. For comparison we would like to refer to Slater, and Arai (1957). Almost the only exception seems to be Green et al. (1953, 1954), where the exact wave function is expanded as a superposition of orthogonal contributions with the HF determinant as its first term ... 

It is instructive to discuss the MP2-R12 method [37] before going into more involved CC-R12. As in MP2, the wave function of MP2-R12 (IT1)) is a linear combination of the reference HF determinant ( o)) and doubly excited determinants produced by the action of a two-electron excitation operator T ... 

Precise and accurate measurements of ultralow Nb, , Zr and Hf concentrations and the chondritic Zr/Hf and Nb/Ta ratios by MC-ICP-MS (Isoprobe, Micromass) were examined by Weyer et al.64 For the development of the isotope dilution technique, enriched 180Ta, 94Zr and 180Hf isotopes were applied for quantitative Zr, and Hf determination, and the monoisotopic element Nb was measured relative to Zr after quantitative separation from the matrix by ion exchange. 

Excited-state energies and wave functions are automatically obtained from Cl calculations. However, the quality of the wave functions is more difficult to achieve. The equivalent of the HF description for the ground state requires an all-singles Cl (SCI). Singly excited configurations do not mix with the HF determinant, that is,... 

If we consider all possible excited configurations that can be generated from the HF determinant, we have a full CI, but such a calculation is typically too demanding to accomplish. However, just as we reduced the scope of CAS calculations by using RAS spaces, what if we were to reduce the CI problem by allowing only a limited number of excitations How many should we include To proceed in evaluating this question, it is helpful to rewrite Eq. (7.1) using a more descriptive notation, i.e.. 

A somewhat special case is the matrix element between the HF determinant and a singly excited CSF. The Condon-Slater rules applied to this situation dictate that... 

As noted in Chapter 6, basis-set flexibility is key to accurately describing the molecular wave function. When methods for including electron correlation are included, this only becomes more true. One can appreciate this in an intuitive fashion from thinking of the correlated wave function as a linear combination of determinants, as expressed in Eq. (7.1). Since the excited determinants necessarily include occupation of orbitals that are virtual in the HF determinant, and since the HF determinant in some sense uses up the best... 

The HF term includes intemuclear repulsions, and the perturbation correction E(2) is a purely electronic term. E1 2 is a sum of terms each of which models the promotion of pairs of electrons. So-called double excitations from occupied to formally unoccupied MOs (virtual MOs) are required by Brillouin s theorem [89], which says, essentially, that a wavefunction based on the HF determinant Z> plus a determinant corresponding to exciting just one electron from Dl cannot improve the energy. 

To construct the HF determinant we used only occupied MOs four electrons require only two spatial component MOs, pi and i//2, and for each of these there are two spin orbitals, created by multiplying ijj by one of the spin functions a or jl the resulting four spin orbitals (i/qa, pi/31p2x, i//2/ ) are used four times, once with each electron. The determinant the HF wavefunction, thus consists of the four lowest-energy spin orbitals it is the simplest representation of the total wavefunction that is antisymmetric and satisfies the Pauli exclusion principle (Section 5.2.2), but as we shall see it is not a complete representation of the total wavefunction. 

The HF determinant A single-excited determinant A doubly-excited determinant... 

MO that resembles an antibonding linear combination of these atomic orbitals. We want these to be our HOMO and LUMO. The CAS wavefunction would then be composed of the HF determinant plus all determinants in which the two formally unpaired electrons are distributed (cf. Fig. 5.2.2) among the HOMO and LUMO. This is the minimum active space for a CAS calculation on these species, and is called a CASSCF(2,2) calculation (2 electrons, 2 MOs). This means that two electrons are being distributed in all possible ways among two MOs. 

The HF (Hartree-Fock) Slater determinant is an inexact representation of the wavefunction because even with an infinitely big basis set it would not account fully for electron correlation (it does account exactly for Pauli repulsion since if two electrons had the same spatial and spin coordinates the determinant would vanish). This is shown by the fact that electron correlation can in principle be handled fully by expressing the wavefunction as a linear combination of the HF determinant plus determinants representing all possible promotions of electrons into virtual orbitals full configuration interaction. Physically, this mathematical construction permits the electrons maximum freedom in avoiding one another. 

As it is not possible to obtain TDDFT-SS results, the results refer to CIS method. In fact, this method can be obtained from two points of view one is to consider the method as a standard Cl, in which the wave function of the excited state is constructed by single excitations from the HF determinant and thus a SS solvent response can be obtained the other is to consider CIS as the result of the Tamm-Dancoff approximation applied to the linear response equation based on the HF wave function. The two ways of looking at the CIS method give the same equations in vacuo, but, as discussed above, they differ for molecules in solution due to the nature of the effective Hamiltonian. 

In principle, a Cl approach provides an exact solution of the many-electron problem. In practice, however, only a finite set of Slater determinants can be handled in the linear expansion. A common procedure is to retain all Slater determinants that differ from the HF determinant by one or two excitations (although one-electron excitations do not couple directly to the ground state they couple with two-electron excitations, which in turn affect the ground state indirectly). Unfortunately, such a procedure is not size consistent. For example, the energy of two highly separated monomers will not be twice that of a single monomer in such a truncated Cl calculation. Fortunately, a slightly modified approach called quadratic Cl has recently been developed (Pople et al., 1987) that is size consistent. 

A high-level calculation based on Hartree-Fock theory in which the HF determinants are mixed so as to describe the dynamic correlation of electrons... 

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