Linear combination of Slater determinants

So, we have learned that a single Slater determinant can adequately describe some electronic configurations, but others can only be described by a linear combination of Slater determinants, even at the lowest level of accuracy. 

Generalization of this one determinant function to linear combinations of Slater determinants, defined for example as these discussed in the previous section 5.2, is also straightforward. The interesting final result concerning m-th order density functions, constructed using Slater determinants as basis sets, appears when obtaining the general structure, which can be attached to these functions, once spinorbitals are described by means of the LCAO approach. 

That is, if we are dealing with non-degenerate states. Otherwise the wave function might be a limited linear combination of Slater determinants. 

The basis of the expansion, ifra, are configuration state functions (CSF), which are linear combinations of Slater determinants that are eigenfunctions of the spin operator and have the correct spatial symmetry and total spin of the electronic state under investigation [42],... 

One of the more radical approximations introduced in the deduction of the Hartree-Fock equations 2 from the Schrodinger equation 3 is the assumption that the wavefunction can be expressed as a single Slater determinant, an antisymmetrized product of molecular orbitals. This is not exact, because the correct wavefunction is in fact a linear combination of Slater determinants, as shown in equation 5, where Di are Slater determinants and c are the coefficients indicating their relative weight in the wavefunction. 

The (V-particle function d>o e La V ) is given, in general as a linear combination of Slater determinants constructed from plane waves, thus extending the treatment of both Macke [53, 54] and of March and Young [55]. Thus, we have = where %k is the Slater determinant xx = (iV) det... 

Eq. (152) describes a non-interacting system. It is clear that is not a subspace of since a linear combination of Slater determinants does not in general yield a single Slater determinant. Applying the same arguments as those of Section 4.1, we conclude that can be decomposed into the orbits (9 ... 

If the r-space wavefunction is a linear combination of Slater determinants constructed from a set of spin-orbitals /. , then its p-space counterpart is the... 

This projection/annihilation approach is probably more useful as an analytical tool, for annihilating the principal spin contaminants from a wave function by hand calculation, for example, than as a computational tool. There is a vast body of literature (see, for example, Pauncz [18]) on generating spin eigenfunctions as linear combinations of Slater determinants, from explicitly precomputed Sanibel coefficients to diagonalizing the matrix of S. However, there are other methods that exploit the group theoretical structure of the problem more effectively, and we shall now turn to these. 

The MCSCF Method. As noted previously, the primary reason for poor convergence in Cl calculations is the use of SCF virtual orbitals in constructing excited configurations these orbitals are not determined variationally and so are rather poor approximations to the true virtual orbitals. The MCSCF method treats a linear combination of Slater determinants Pi,... 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

Each such Lewis structure is represented by a single VB spin-eigenfunction (HVH n), hereafter called a "VB structure". These VB structures are linear combinations of Slater determinants involving the same occupied AOs as the corresponding Lewis structures, as in eqs 10-12. 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

The total, antisymmetric function for a closed-shell configuration is expressed as a Slater determinant built-up from the spin-orbitals. In the case of open-shell configurations, a linear combination of Slater determinants may be needed in order to obtain a function with the same symmetry and multiplicity characteristics as the state under consideration. 

As we have seen previously (Chapter 5), the eigenfunction for a polyelec-tronic atom is antisymmetric with respect to the exchange of the coordinates of any two electrons, and can be expressed as a Slater determinant whose elements are the various occupied spin-orbitals (or a linear combination of Slater determinants, in the case of open-shell atoms). The same appfies to polyelectronic molecules, the atomic orbitals being replaced by the various occupied molecular orbitals associated with the a and /3 spin-functions spin molecular orbitals. Thus, for the molecules H2O, NH3 or CH4 having five doubly occupied m.o.s (one core s orbital and four valence m.o.s), we have... 

In the second step which is a core part of the DV-ME method, the many-electron wave functions are described as linear combinations of Slater determinants, and all the matrix elements of the many-electron Hamiltonian are calculated, then finally diagonalized to obtain the multiplet energies and many-electron wave functions. 

Next are methods based on zeroth-order wave functions that are linear combinations of Slater determinants that again may or may not include a treatment of residual electron correlation effects. In these multiconfigurational (MC) approaches, the starting wave function is presupposed to be given by a determinantal expansion... 

The MODPOT/VRDDO LCA0-M0-SCF programs have been meshed in with the configuration interaction programs we use. In this Cl (71) program each configuration is a spin- and symmetry-adapted linear combination of Slater determinants in the terms of the spin-bonded functions of Boys and Reeves ( 2, 3> 7 0 as formu-... 

According to Lykos and Parr 3>, an unsaturated molecule can be described in this manner by taking for S a (in principle complete) linear combination of Slater determinants built from a orbitals only and for n a similar combination built from n orbitals only. Such a description is more general than the independent-particle model, as it includes the latter as a special case (namely where both and 77 are single Slater determinants). 

If a -electron wave function is limited to a Slater determinant of n spin orbitals, one stays within the frame of the independent-particle model, and the best model of that sort (for a discussion, see 22>) for a given problem is that in which the orbitals used to construct the wave function are solutions of the Hartree-Fock equations. This model is only an approximation of the correct wave function. As mentioned in Sect. 3.1, the wave function should be written as a linear combination of Slater determinants, as in Eq. (3.4). To illustrate this, let us consider a two-electron system where the spin can be separated off, so that it is sufficient to consider a function ip (1,2) depending only on the space coordinates of the two particles 1 and 2. For a singlet state ip (1,2) is symmetric with respect to space coordinates ... 

The correlation problem can be solved in principle by configuration interaction (Cl) or one of its variants. The exact wavefunction is expanded as a linear combination of Slater determinants ... 

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