Orthogonalization Slater determinants

The primary feature of SCVB is the use of non-orthogonal orbitals, which allows a much more compact representation of the wave function. An MO-CI wave function of a certain quality may involve many thousand Slater determinants, while a similar quality VB wave function may be written as only a handful of resonating VB structures. 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), <, which are orthogonal (Section 4.2). 

With this choice for H°, equations (7) and (8) are automatically valid for the perturbation. The only restriction is that we have to use orthogonal orbitals and Slater determinants rather than Configuration State Functions (CSFs) as a basis for the perturbation. None of these restrictions is constraining, however. 

The same expression can be used with the appropriate restrictions to obtain matrix elements over Slater determinants made from non-orthogonal one-electron functions. The logical Kronecker delta expression, appearing in equation (15) as defined in (16)] must he substituted by a product of overlap integrals between the involved spinorbitals. 

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. 

Recently, an alternative scheme based on singlet-type strongly orthogonal geminals (SSG) was proposed [5]. In this scheme, the wavefunction is split into gem-inal subspaces depending on the number of spin-up or spin-down electrons, n and n, respectively, while the wavefunction is filled up with one Slater determinant. 

Note that the additional normalization factor det T appears because the cpA and Slater determinant are not orthogonal therefore, lelectron density of lcpAayBal is not equal to simply the sum of the orbital densities. Since in the Slater determinant lygayual the orbitals yg and yu are orthogonal, the density is the sum lygl2 + lyj2, which is easily seen to be just the p° of Eq. [12]. 

This technique utilizes an option, offered by most ab initio standard programs, to compute the energy of any guess function even if the latter is based on nonorthogonal orbitals. The technique orthogonalizes the orbitals without changing the Slater determinant, and then computes the expectation... 

To illustrate the modifications of UHF formalism, it is convenient to consider pure spin symmetry for a single Slater determinant with Nc doubly occupied spatial orbitals Xi and N0 singly occupied orbitals y". The corresponding UHF state has Na mj = occupied spin orbitals and Np rns = — J, occupied spin orbitals f. The number of open-shell and closed-shell orbitals are, respectively Na = Na — Np > 0 and Nc = Np. Occupation numbers for the spatial orbitals are nc = 2, n ° = 1. If all orbital functions are normalized, a canonical form of the RHF reference state is defined by orthogonalizing the closed- and open-shell sets separately. 

The detailed study of electron distribution rearrangements in a chemical reaction needs a multiconfiguration ab initio method, because one determinantal wavefunctions are unreliable away from equilibrium geometries. By means of the CASSCF method it is possible to focus attention on the electrons more directly involved in the reaction, allowing the calculation to be done with a relatively limited number of Slater determinants. Moreover, CASSCF uses orthogonal orbitals which are simpler than non-orthogonal orbitals in the development of computer codes. Nowadays CASSCF is, in fact, efficiently included in practically all distributed packages for molecular quantum calculations. 

Firstly, the function (70) is invariant under a linear transformation of the m doubly occupied orbitals amongst themselves. A proof of this statement seems hardly necessary as, in the case m = N, equation (70) is equivalent to a Slater determinant, and this property of a determinant is well-known. The m orbitals m may therefore be orthogonalized amongst themselves by a linear transformation, without altering the total wavefunction. This, of course, may be done in several ways, by transforming to MOs for example, but perhaps the most convenient method is to employ the Lowdin symmetric orthogonalization method 73... 

Slater determinants constructed from non-orthogonal spin orbitals. If we now denote the matrix of overlap integrals between spin orbitals... 

It is also of interest to examine the dissociation of this model molecule A-B into (one-electron) atoms A and B, which determine the promolecule. Such decoupled AO corresponds to the molecular configuration [< ond vlna] since the Slater determinant l wi. = ab. Indeed, using the orthogonal transformations between / = (a, b) and

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