Determinantal method

Slater developed an approach (the "determinantal method") that offers a way of choosing among linear combinations (essentially sums and differences) of the polar and nonpolar terms in the Hund-Mulliken equations to bring their method into better harmony with the nonpolar emphasis characteristic of the Heitler-London-Pauling approach in which polar terms do not figure in the wave equation. 72... 

At this point some misleading nomenclature in publications of simulations has to be clarified. Frequently simulations based on DFT functionals are termed ab initio MD and first principle or -empirical. This is indeed not correct as all available density functionals are of a more or less semiempirical nature, as pointed out by their developers themselves, e.g., by Becke in his presentation of the B3LYP functional (15). The term ab initio should be reserved, therefore, for simulations where for the QM part a true ab initio procedure, i.e., HF or correlated methods like MP/2 or better, is employed. Only by this one is also enabled to perform a method-inherent control of accuracy and deficiencies by increasing the level of theory from a one-determinantal to a multi-determinantal method. 

The original Heitler-London calculation, being for two electrons, did not require any complicated spin and antisymmetrization considerations. It merely used the familiar rules that the spatial part of two-electron wave functions are symmetric in their coordinates for singlet states and antisymmetric for triplet states. Within a short time, however, Slater[10] had invented his determinantal method, and two approaches arose to deal with the twin problems of antisymmetrization and spin state generation. When one is constructing trial wave functions for variational calculations the question arises as to which of the two requirements is to be applied first, antisymmetrization or spin eigenfunction. 

In 1929 (Phys. Rev., 34, 1203) John Slater published his famous (non-group-theoretical) determinantal method for constructing antisymmetric fermion functions. In 1927 (Proc. Roy. Soc., 114A 243) Paul Dirac introduced the... 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

Although in the case of f2, either method can be applied with ease, the determinantal method becomes cumbersome for more than two-electrons and it is necessary to take recourse to the tensor operator method for fN, N > 2. We will show the derivation of energy levels by both methods in what follows ... 

Slater determinantal method Part of the microstate table for f2 is given in Table 8.32. 

These energies are same as those derived by the Slater determinantal method. 

Note that the above energies are actually the diagonal elements. In order to calculate the off-diagonals, it is necessary to take recourse to either the Slater determinantal method or the tensor operator method. 

Determinantal method In this method it is necessary to calculate the one electron matrix elements of an f electron. These are, in terms of tensor operator parameters... 

These are identical to the matrix elements obtained by the determinantal method. 

In using the determinantal method, the explicit use of a spin coordinate and spin-orbitals for each electron has provided the simplest mathematically elementary technique for ensuring that this permutational symmetry is obeyed and retained. No new algebraic or analytical tools were required the definition of spin integration is all that is needed. But the price which had to be paid for this mathematical simplicity was the entanglement of space emd spin variables. It would be physically and chemically attractive to be able to remove any explicit appearance of electron spin from the valence bond model and concentrate on the main physical features of the model which are ... 

With this in mind it is worth asking how practical the expansion of a wave-function in terms of a large number of determinants actually is. This question can only be answered by carrying out detailed calculations but if we try to interpret the mathematics of the determinantal method we may get a clue. 

M. Sironi, D. L. Cooper, M. Raimondi, Valence Bond Theory Determinantal Methods. In Handbook of Molecular Physics and Quantum Chemistry, volume 2 Molecular Electronic Structure, chapter 11, p. 140, S. Wilson, R F. Bernath, R. McWeeny, eds., John Wiley, Chichester (2003)... 

TD-DFT is a mono-determinantal method, and thus it cannot be applied to electronic states with an intrinsic multireference character [2]. Analogously, TD-DFT can exhibit deficiencies in treating electronic transitions with substantial contributions from double excitations [63-66], although interesting attempts to overcome the above limitations have been proposed [65-71]. In several cases, however, an electronic transition exhibits a multireference character (or a significant contribution from double excitations) just because of a poor description of the ground-state MO by HF orbitals, and such features are not present when using MOs computed at the DFT level. 

And finally, the determinantal method permits one to be closer to the physical interaction than the tensorial way. The levels are only abstractions whereas all the known interactions are defined as one-, two- or three-electron operators (l/ri2. I s,...). 

Virtually all of the successful path integral simulations of 2-d models for electronic systems have been carried out by the auxiliary field MC method, sometimes called the determinantal method. The only thing that complicates the computation of the fermion partition in equation (8) is the interaction action 5i. As explained in Section 5.3, without 5i, the sum over exchanges can be performed analytically. Therefore, if the two-electron interaction term can be eliminated or at least decoupled, the fermion sign problem could be partially removed. This can be accomplished by a so-called Hubbard-Stratonovich transformation. The details can be found in the original paper. Briefly, two electrons (of opposite spin) on the same site i experience a repulsion of strength U and add a term —eUni ni to the action Si, where = 0, 1 is the occupation number of an f-spin electron on site i, and n, is the same for a -spin electron. To decouple the two-electron interaction, the following transformation (correct up to a multiplicative constant) can be used. 

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