Spin-density matrices

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

Spin-dependent operators are now introduced. The external potential can be an operator Vext acting on the two-component spinors. The exchange-correlation potential is defined as in Eq. [27], although Exc is now a functional Exc = Exc[pap] of the spin-density matrix. The exchange-correlation potential is then... 

When adding the spin density matrices P and P3 to yield the total density matrix P, P = P + P3 and the total spin density matrix Ps being the difference of P and P 3, Ps = P - P3, we... 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

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. 

C. Spin density matrix of a spin system at thermal equilibrium with a lattice. 231... 

C. Evaluation of a spin density matrix after exchange.244... 

The spin density matrix Pj(t) which describes the properties of any spin system of a molecule A, is defined as follows. We assume that the density matrices Pj(0), j = 1, 2,..., S, which describe the individual components of the dynamic equilibrium at any arbitrary time zero, are known explicitly, and that at any time t such that t > t > 0 the pj(t ) matrices are already defined. Our reasoning is applied to a pulse-type NMR experiment, and we therefore construct the equation of motion in a static magnetic field. The p,(t) matrix is the weighted average over the states involved, according to equation (5). The state of a molecule A, formed at the moment t and persisting as such until t, is given by the solution of equation (35) with the super-Hamiltonian H° ... 

Here P is a spin-density matrix, defined as a difference of density matrices for a and p spin, the Pa, Pe. Molecular orbitals F are approximated by linear combinations of atomic orbitals Xi> the other symbols have their usual meanings. 

Dyakonov, M.I. and Perel, V.I. (1972). General inequalities for the relaxation constants of a spin density matrix, Phys. Lett., 41 A, 451-452. 

Cl calculation. In addition it should include important single excitations. Because the coefficients of the single excitations are rather small, the process of selecting important single excitations should include an analysis of the spin density matrix of a foregoing Cl calculation. 

The system is described by a spin density matrix, p(t). The electron spins are then allowed to evolve under the spin Hamiltonian (Equation 8.13 with the exchange term removed) by application of the Liouville-Von Neumann equahon. 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

The computational procedure follows closely the steps of an actual m.p. experiment see Fig. 1. The spin system, which is initially in thermal equilibrium, is hit by a preparation pulse Pp. Thereafter, one component of the transverse nuclear magnetization created by Pp, say My, is measured and the measurement is repeated at intervals of the cycle time The resulting time series My(qtJ,q = 0,...,(2 " - 1), if Fourier transformed. For simulations we accordingly first specify the initial condition of the spin system, that is, the initial value of the spin density matrix g(t) in the rotating frame. Our standard choice Pp, = P implies p(0) fy == the sum running over k = We then follow the evolution... 

The spin density matrix, p =< c c > is defined in terms of the coefficients of the spin states, ip = Cnipn-... 

In an external magnetic field, the dynamics of these long-lived coherences are superposed with those generated by the spin-dependent interactions. In the context of NMR line shape experiments, the overall evolution of the relevant spin-space coherences can be described in the sole spin manifold the final DQR equation for the effective spin density matrix of a methyl-like rotor reads... 

The spin dependent Wigner functions can be obtained either from the spin density matrix or directly from a spin constracted Wigner function as discussed in [10], which we have used in the past. We deal with a modified version of the Wyle [19] transformation ... 

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