Expansion in Slater determinants

The approximate methods differ then in how the energy is calculated and how the molecular orbital coefficients, c p, and the coefficients C o in the expansion in Slater determinants are determined. In this context, we can distinguish between... 

It is now clear that, as in the direct method of expansion in Slater determinants (p. 239), the practical feasibility of calculations by this method will depend heavily on the availability of efficient algorithms for the computation of cofactors. This problem, which we do not consider further, has been studied in particular by King et al. (1967), Prosser and Hagstrom (1968), and again by Gallup et al. (1982). 

The only method found so far which is flexible enough to yield ground and excited state wavefunctions, transition rates and other properties is based on expanding all wavefunctions and operators in a finite discrete set of basis functions. That is, a set of one-particle spin-orbitals < >. s-x are selected and the wavefunction is expanded in Slater determinants based on these orbitals. A direct expansion would require writing F as... 

A term in the above expansion corresponds to a closed path of P steps in Slater determinant space ... 

We begin our discussion of wave function based quantum chemistry by introducing the concepts of -electron and one-electron expansions. First, in Sec. 2.1, we consider the expansion of the approximate wave function in Slater determinants of spin orbitals. Next, we introduce in Sec. 2.2 the one-electron Gaussian functions (basis functions) in terms of which the molecular spin orbitals are usually constructed the standard basis sets of Gaussian functions are finally briefly reviewed in Sec. 2.3. 

The second approach relies on the expansion of the SC wave function in Slater determinants constructed from non-orthogonal orbitals. The density matrix is calculated from cofactors, constructed in situ using graphical indexing techniques. Calculations have been carried out with this strategy for up to 14 active electrons. It has also been extended recently to include simultaneous optimization of the inactive electrons. ... 

In the CI expansion (11.1.1), the basis functions i) are Slater determinants. As discussed in Section 2.5, a more compact representation is obtained by expanding the wave function in the spin-symmetrized CSFs rather than in Slater determinants. Still, in the present chapter, we shall always assume that the CI wave function is expanded in the sin ler basis of Slater determinants. As discussed in Section 2.6.6, the linear transformation between these two alternative representations of the CI wave function can be carried out rather easily. We may therefore adapt the determinantal techniques of the present chapter to CSF expansions simply by expanding the CSFs in determinants just before any calculation or manipulation is to be carried out on the CI wave function and transforming back again immediately afterwards. Alternatively, we may employ any of the many special techniques that have been developed for performing the computational tasks of CI theory directly in the CSF basis - without recourse to the basis of Slater determinants. However, for high efficiency and generality, the simpler determinantal basis is more useful. 

Once our N-electron CSF has been reduced to a linear combination of one or two (N — 1)-electron spin eigenfunctions each multiplied by a creation operator, we may go one step further and expand each (N — l)-electron state in two (N — 2)-electron spin eigenfunctions as dictated by the penultimate element t -i in the genealogical coupling vector t. After N — I such steps, we arrive at an expansion in terms of determinants with projected spin M. In this way, we are led directly to an expansion of CSFs in Slater determinants where the coefficients are products of the genealogical coupling coefficients in (2.6.5) and (2.6.6). 

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. 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

Moreover, there are 2 terms in the expansion of the Slater determinant for He but there are 6 terms for Li. Looking at beryllium, we find 24 terms. This is the beginning of the faetorial series... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

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. 

Iosio Kato in 1957. [92] Unfortunately, any trial wave function composed of Slater determinants has smooth first and higher derivatives with respect to the interelec-tronic coordinates. Thus, even though such expansions are insightful and preserve the concept of orbitals to some extent, from a mathematical point of view they are expected to be slowly convergent. 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

Unlike the density cumulant expansion, which can in principle be exact for certain states (such as Slater determinants), the operator cumulant expansion is never exact, in the sense that we cannot reproduce the full spectrum of a three-particle operator faithfully by an operator of reduced particle rank. However, if the density cumulant expansion is good for the state of interest, we expect the operator cumulant expansion to also be good for that state and also for states nearby. 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

The primary challenge in quantum chemistry is to find a good approximation to the electronic wave function of a quantum state. We can express any N-electron wave function in a complete basis of Slater determinants, through the FCI expansion,... 

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