Slater determinant linear combination

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

Slater showed that spinorbitals, arrayed as a determinant, change sign on election exchange so as to obey the Pauli principle. If we wi ite a linear combination of two spinorbitals as a determinant where we assume the space parts are the same but the spin parts are not the same... 

HyperChem uses single determinant rather than spin-adapted wave functions to form a basis set for the wave functions in a configuration interaction expansion. That is, HyperChem expands a Cl wave function, in a linear combination of single Slater determ in ants... 

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. 

As computational facilities improve, electronic wavefunctions tend to become more and more complicated. A configuration interaction (Cl) calculation on a medium-sized molecule might be a linear combination of a million Slater determinants, and it is very easy to lose sight of the chemistry and the chemical intuition , to say nothing of the visualization of the results. Such wavefunctions seem to give no simple physical picture of the electron distribution, and so we must seek to find ways of extracting information that is chemically useful. 

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... 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

The HPHF wavefunction for an 2n electron system, in a gronnd state of S qnantum number, even or odd, is written as a linear combination of only two DODS Slater determinants, built up with spinorbitals which minimize the total energy [1-2] ... 

Also, an alternative formulation of equation (17) can be conceived if one wants to distinguish between ground state, monoexcitations, biexcitations,. .. and so on. Such a possibility is symbolized in the following Cl wavefunction expression for n electrons, constructed as to include Slater determinants up to the p-th (pp) unoccupied ones l9klk=i,ni Then, the Cl wavefunction is written in this case as the linear combination ... 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

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. 

Strictly speaking, this statement applies only to closed-shell systems of non-degenerate point group symme-try, otherwise the wave function consists of a linear combination of a few Slater determinants. 

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],... 

The individual J,Mj) multiplets cannot be, in principle, described by a single Slater determinant. This is a consequence of the general rules of coupling of two angular momenta eigenfunctions of all projections of the total angular momentum which is lower than the maximal possible (L + S), are represented as linear combinations of several determinants [36]. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

In analogy to using a linear combination of atomic orbitals to form MOs, a variational procedure is used to construct many-electron wavefunctions from a set of N Slater determinants y, i.e. one sets up a N x. N matrix of elements flij = (d>, H d>y) which, upon diagonalization, yields state energies and associated vectors of coefficients a used to define (fi as a linear combination of A,s ... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

The Kohn-Sham equations can be obtained from the minimalization of the noninteracting kinetic energy after expressing it with one-electron orbitals. Because Tl 0 is generally a linear combination of several Slater determinants, the form of the... 

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 ... 

Eq. (31) is a convex linear combination of 2-densities from Slater determinants. Geometrically, is a polyhedron and its vertices correspond to pure states. 

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