Operators Antisymmetric

In particular, within the orbital model of eleetronie strueture (whieh is developed more systematieally in Seetion 6), one ean not eonstruet trial waveflmetions whieh are simple spin-orbital produets (i.e., an orbital multiplied by an a or P spin funetion for eaeh eleetron) sueh as lsalsP2sa2sP2pia2poa. Sueh spin-orbital produet funetions must be made permutationally antisymmetrie if the N-eleetron trial funetion is to be properly antisymmetrie. This ean be aeeomplished for any sueh produet wavefunetion by applying the following antisymmetrizer operator ... 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-eleetron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Zp Sp P eommute with total =Zi (i), as in the linear-moleeule ease. The additional symmetry present in the spherieal atom refleets itself in the faet that Lx, and Ly now also eommute with H and A. However, sinee does not eommute with Lx or Ly, new quantum... 

The 1 operator is the identity, while Py generates all possible permutations of two electron coordinates, Pyi all possible permutations of three electron coordinates etc. It may be shown that the antisymmetrizing operator A commutes with H, and that A acting twice gives the same as A acting once, multiplied by the square root of N factorial. 

In addition to the Schrodinger equation we have the antisymmetry requirement (Eq. II.2) connected with the Pauli principle and, by means of the antisymmetrization operator (Eq. 11.16), the Hartree product (Eq. 11.37) is then transformed into a Slater determinant ... 

The AB supermolecule is described by a single determinant wave function formulated in terms of doubly occupied molecular orbitals with no orthonormality constraints. For a system with 2N = 2Na +2Nb electrons the SCF-MI wave function expressed in terms of the antisymmetrizer operator A is... 

The requirement of overall exchange antisymmetry of the /V-clcct.ron wavefunction [Pg.36]

In Eq. (1.16a), A is the antisymmetrizer operator that converts the orbital product into a Slater determinant, insuring satisfaction of the Pauli exclusion principle. In this equation (alone), the same spatial orbital might appear twice, with different indices to indicate the change in spin. For example, / i (0,(7 ypf HA) might be the same as i<0)(F K/>,0,0" 2). a doubly occupied spatial orbital (n]m> = 2), with a bar denoting opposite spin in the second spin-orbital. 

Asm is an antisymmetrizer operator between electrons from these two groups s and m which is usually expressed as a sum of the identity operator (1) and normalized permuting operator Pms Asm =l+pms. The total Hamiltonian is symmetric to any electron permutation. The interaction energy Vsm can be cast in terms of a direct Coulomb interaction and an exchange Coulomb interaction ... 

A tableau may be used to define certain subgroups of which are themselves direct products of smaller permutation groups the symmetrizing and antisymmetrizing operators for these subgroups lead, as we shall see, to projection operators on irreducible representations of... 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

The last equality holds since 0 is normalized and will be constrained to remain so upon variation of the orbitals. Before we substitute equation (A. 10) into equation (A.14), a simplifying observation can be made. Equation (A. 10) [and (A. 12)] can be expressed in terms of the antisymmetrizer operator, A,... 

Our first objective is to derive a simpler expression for the electronic energy. We can do this by using the properties of the antisymmetrizer operator indicated at the right. Thus,... 

Comment. It is the action of the antisymmetrizing operator on the product state function in eq. (7.2.5) that produces the Slater determinant in eq. (7.2.6). However, the factorization into an orbital function of r and a spinor, that simplifies our work when N 2, does not occur for N>2.)... 

When the two systems A and B are brought from infinity to their equilibrium postions, the wavefunctions FA and TB of the subsystems will be overlapping. The Pauli principle is obeyed by explicitly antisymmetrizing (operator A) and renormalizing (factor N) the product wavefunction ... 

This problem is solved by explicitly including the antisymmetrization operator in the definition of the group function [45] ... 

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