Density matrix unrestricted

The challenge with unrestricted methods is tire simultaneous minimization of spin contamination and accurate prediction of spin polarization . The projected UHF (PUHF, see Appendix C) spin density matrix catr be employed in Eq. (9.36), usually with somewhat improved results. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

The unrestricted Hartree-Fock (UHF) case is completely analogous to the closed-shell one. New terms do appear in the open-shell SCF and the few-configuration case. Nevertheless, the preferred technique is quite similar to the closed-shell case. In particular, the two-particle density matrix can be constructed from compact matrices, and the solution of the derivative Cl equations is very simple, due to the small dimension. 

The unrestricted and restricted open-sheU Hartree-Fock Methods (UHF and ROHF) for crystals use a single-determinant wavefunction of type (4.40) introduced for molecules. The differences appearing are common with those examined for the RHF LCAO method use of Bloch functions for crystalline orbitals, the dependence of the Fock matrix elements on the lattice sums over the direct lattice and the Brillouin-zone summation in the density matrix calculation. The use of one-determinant approaches is the only possibility of the first-principles wavefunction-based calculations for crystals as the many-determinant wavefunction approach (used for molecules) is practically unrealizable for the periodic systems. The UHF LCAO method allowed calculation of the bulk properties of different transition-metal compounds (oxides, perovskites) the qrstems with open shells due to the transition-metal atom. We discuss the results of these calculations in Chap. 9. The point defects in crystals in many cases form the open-sheU systems and also are interesting objects for UHF LCAO calculations (see Chap. 10). 

The density matrix idempotency relations described here may be easily extended to the unrestricted Hartree-Fock (UHF) method when the orbitals for a and / spins are treated independently. 

Unpaired electron spin density p. Negative values of p for each H nucleus of PH2 (and thus for the related isotropic H magnetic hf coupling constant ap( H) see p. 65) are to be expected in view of a spin-polarization effect in the P-H o bond [5 to 7]. Numerical data were derived from ap( H) from two ESR spectra p = (-)0.035 (Kr matrix) [5] and -0.046 (cancrinite matrix) [6]. The following spin densities were theoretically calculated by ab initio (unrestricted Hartree-Fock, UHF) and semiempirical (INDO) methods [8] ... 

The situation is more complicated if the restriction of double occupancy is dropped. Here we discuss the unrestricted Hartree-Fock (UHF) method in some detail. In this case one has different density matrices for the a and p electrons, P and pP, respectively. As a consequence, one has also different Fockians for the a and P electrons F and F. The expressions of their matrix elements can be derived from Eq. (10.55) by performing the integration over spin functions ... 

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