Density matrices spinless

For many purposes we need the spinless density matrices />(1 10 = y(l l )dsi... 

Similarly as with the overlap determinant method the primes are used to denote the fact that the AO bases x and % serving to describe the molecular orbitals of the reactant and the product are generally different. Using the known expressions for the first order spinless density matrices (136), the original definition expression for the similarity index (137) can be rewritten in the form (138). 

When the integrals are defined over occupied orbitals the A and B coefficients are elements of the spinless density matrices P and /7 but if instead we define the integrals with respect to. ypm-orbitals then the appropriate coefficients will be elements of the density matrices p and n (includmg spin). Thus both variation problems may be discussed using the same formalism, merely by changing the interpretation of the integrals and their coefficients. In both cases, the A and B coefficients are defined in terms of similar (transition) quantities for all pairs of CFs i) in the expansion (8.2.1) for example. 

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 ... 

Consider the operator H of the crystal-plus-impurity system let us denote by i, ) and its eigenstates and eigenfimctions. The (spinless) density matrix operator for the ground state is... 

If we are considering the two-particle matrix the obvious thing to do is to expand in a basis consisting of at least the two atomic orbitals 9 and on the two sites of attack, and look for the coefficient (C b) in the spinless density matrix of the form 9 a(1) 9b(2) 95a(10 b(2 ). The appropriate term in the one-particle matrix would then. be [Pg.312]

Let us introduce the spinless density matrix P referring to the spatial orbitals as ... 

Note that, for the convenience of later discussion, we define the density operator and density matrix as the corresponding quantities of either the spin-up or spin-down component for a spin-compensated system. It differs from the definition of the usual spinless density matrix by a factor of 1/2. ... 

In practice, we are usually concerned with the orbital space spanned by a set of m AOs, or other basis functions, spin integrations having been performed already. The spinless density matrix in the closed-shell form (6.2.77) is not itself idempotent, but on writing... 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

The SLG energy can be rewritten as a function of the intrabond matrix elements of spinless one- and two-electron density matrices. However, some regrouping of terms makes the picture more clear. Using the above expressions for the density matrix elements, one can easily write the equation for electronic energy. The contributions from the one-center terms (T nA < >) have the form ... 

The electron density (10) is the so-called diagonal element of a more general quantity, the (spinless) one-electron density matrix, P(r, r ), defined in exactly the same way except that the variables in it carry primes - which are removed before the integrations. The reduction to (11), in terms of a basis set, remains valid, with a prime added to the variable in the starred function. For a separable wavefunction, the density matrices for the whole system may be expressed in terms of those for the separate electron groups in particular, for a core-valence separation,... 

Yrs are two-centre two-electron repulsion energies. For aromatic ions, the onsite electron pair densities, i.e., the diagonal elements of the spinless second order density matrix are lower than those of any classical structure. For radicals, however, the pair densities are increased relative to those in the corresponding classical structures. Thus the resonance energy of the ion exceeds that of the radical by 21 Ec I [33]. FIMO and PMO do not differentiate between the formation of an arylmethyl radical and its carbocation. Empirically, radicals are better described by these methods this has been related to the constant Coulson charge order Q = 1 for arylmethyl radicals as opposed to the variable rt-charge order 1 on arylmethyl cations [16,39]. The actual PPP values of Ec have been correlated to an excellent accuracy to their PMO-o) counterparts [16]. 

The first property of interest is the spinless single-particle density matrix y(r, r ) defined as... 

The field F(r) (see Eq. 59) depends on the wavefunction 4 through the density p(r), spinless single-particle density matrix y(r, r ), and the pair-correlation density g(r, r ). Furthermore, this work is path-independent since the field F(r) is conservative. The path of the inverse map C, whereby for every ground-state wavefunction P there corresponds a potential v(r), is now well defined. 

In the Hartree-Fock operator F the term Hq is the one-electron operator (4.8). The action on the function v <(r) of the Coulomb operator J and exchange operator K is determined in the following way. Denote by p r,r ) the mixed electron density with fixed spin (spinless electron-density matrix)... 

The spinless one-electron density matrix (DM) elements are defined in the LCAO approximation as... 

As is well known, the energy of a system within the single-determinant Hartree-Fock approximation can be expressed in terms of the one-electron density matrix (DM). The one-electron spinless DM p R, R )is defined as... 

The orbital defined as the eigenfunction of the spinless one-particle electron density matrix. See Population Analyses for Semiempirical Methods. 

Use the result obtained in Problem 14.2 to verify the forms of the 1- and 2-electron densities and transition densities (p, Jt) as given (14.1.6). Show that when, 0 are singlet functions the corresponding spinless densities (P, II) follow easily. [Hint Compare your expressions with (5.4.3) and (5.4.4), which give the general matrix elements in terms of denrity functions. Remember that p has only two components when S = 0.j... 

The matrix element (H d ) is determined according to (5.4.3) and (5.4.4) in terms of 1- and 2-electron transition densities, and may be obtained in a precisely similar manner (McWeeny and Sutcliffe, 1963). Since, however, it is usual to employ a spinless Hamiltonian, further reductions will be necessary, and these are less straightforward. Two new factors must be recognized (i) even when the functions, 0 are spatially non-degenerate, they may appear in spin-degenerate sets (e.g. triplets, quintuplets,.. . ), and the excited functions for AB will then be obtained by vector coupling -, and (ii) when non-singlet states occur the further reduction will also lead to spin densities. 

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