Antisymmetric/antisymmetries problem

Clearly, standard Rayleigh-Schrodinger perturbation theory is not applicable and other perturbation methods have to be devised. Excellent surveys of the large and confusing variety of methods, usually called exchange perturbation theories , that have been developed are available [28, 65]. Here it is sufficient to note that the methods can be classified as either symmetric or symmetry-adapted . Symmetric methods start with antisymmetrized product functions in zeroth order and deal with the non-orthogonality problem in various ways. Symmetry-adapted methods start with non-antisymmetrized product functions and deal with the antisymmetry problem in some other way, such as antisymmetrization at each order of perturbation theory. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

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