Spin and Permutation Symmetry

It has been demonstrated that a given eleetronie eonfiguration ean yield several spaee- and spin- adapted determinental wavefunetions sueh funetions are referred to as eonfiguration state funetions (CSFs). These CSF wavefunetions are not the exaet eigenfunetions of the many-eleetron Hamiltonian, H they are simply funetions whieh possess the spaee, spin, and permutational symmetry of the exaet eigenstates. As sueh, they eomprise an aeeeptable set of funetions to use in, for example, a linear variational treatment of the true states. 

We mentioned earlier that the dimensionality of the FCI space is significantly reduced due to spin symmetry. This can be formulated somewhat differently due to the relation existing between the spin and permutation symmetries of the many-electronic wave functions (see [30,42]). Indeed, the wave function of two electrons in two orbitals a and b allows for six different Slater determinants... 

The presence of spin in the trial function does not modify the previous point group symmetry discussion. However the requirements of spin and permutational symmetry may mean that a trial function is constrained so that it might not be possible to have a trial function with a particular point group symmetry and a specified spin symmetry. As an example consider a determinant of doubly occupied orbitals for four electrons symbolized as... 

SPIN AND PERMUTATION SYMMETRY matrices must assume the block form... 

Spin multiplicity, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 706-711 Spin-orbit coupling conical intersections ... 

Every ket in FSP([ASP]) transforms as the [ASP]th irreducible representation of 5 p. A state described by a ket contained entirely within one, and only one, of these subspaces FSP([ASP]) is said to be a pure permutation state and to possess the spin-free permutational symmetry [ASP],... 

The use of the conventional spin formulation in conjunction with a spin-free Hamiltonian HSF merely assures symmetry adaptation to a given spin-free permutational symmetry [Asp] without recourse to group theory. In fact, one may symmetry adapt to a given spin-free permutational symmetry without recourse to spin. This is the motivation behind the Spin-Free Quantum Chemistry series.107-116 In this spin-free formulation one uses a spatial electronic ket which is symmetry adapted to a given spin-free permutational symmetry by the application of an appropriate projector. The Pauli-allowed partitions are given by eq. (2-12) and the correspondence with spin by eqs. (2-14) and (2-15). Finally, since in this formulation [Asp] is the only type of permutational symmetry involved, we suppress the superscript SF on [Asp],... 

Spin-free permutational symmetry is conserved, and because of the correspondence eq. (2-14) between spin S and [A], spin is also conserved. We denote this result... 

The present work details the derivation of a full coupled-cluster model, including single, double, and triple excitation operators. Second quantization and time-independent diagrams are used to facilitate the derivation the treatment of (diagram) degeneracy and permutational symmetry is adapted from time-dependent methods. Implicit formulas are presented in terms of products of one- and two-electron integrals, over (molecular) spin-orbitals and cluster coefficients. Final formulas are obtained that restrict random access requirements to rank 2 modified integrals and sequential access requirements to the rank 3 cluster coefficients. 

PERMUTATIONAL SYMMETRY AND THE ROLE OF NUCLEAR SPIN IN THE VIBRATIONAL SPECTRA OF MOLECULES IN DOUBLY DEGENERATE ELECTRONIC STATES ... 

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

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