Second-order electron density

The second order electronic density matrix can be constructed from the Hamiltonian matrix eigenvector given in eq(5) above to give... 

As discussed earlier, we cannot only derive first-order electron densities, but also we can extend them to higher order electron densities. We have used the second-order electron density p(ri,r2) in lieu of the first-order electron density on several occasions in molecular quantum similarity, because the second-order electron density is in fact the lowest order density where electron correlation becomes apparent. It has been used extensively by Ponec et al. "- in the study of similarity in pericyclic reactions where the second-order electron density offers important advantages over the first-order electron density. In another contribution, Ponec et al. went to the third-order electron density. Again, most of the discussion relating to molecular quantum similarity indices and molecular alignment is also applicable to higher order electron densities, replacing where necessary the first-order electron density by, for example, the second-order electron density. 

Hamiltonians involving more than two electron interactions. I shall use this to illustrate the general case of arbitrary p. The second-order reduced density matrix (2-RDM) of a pure state ij/, a function of four particles, is defined as follows ... 

P. W. Ayers and M. Levy, Generalized density-functional theory conquering the A-representability problem with exact functionals for the electron pair density and the second-order reduced density matrix. J. Chem. Set 117, 507-514 (2005). 

The formulas above give the gradient and the Hessian in terms of matrix elements of the excitation operators. They can be evaluated in terms of one-and two-electron integrals, and first and second order reduced density matrices, by inserting the Hamiltonian (3 24) into equations (4 9), (4 11), and (4 13)-(4 15). Note that transition density matrices and are needed for the evaluation of the Cl coupling matrix (4 15). 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

J. Cioslowski, J.V. Ortiz, One-electron density matrices and energy gradients in second-order electron propagator theory. J. Chem. Phys. 96, 8379-8389 (1992)... 

The RASSI method can be used to compute first and second order transition densities and can thus also be used to set up an Hamiltonian in a basis of RASSCF wave function with separately optimized MOs. Such calculations have, for example, been found to be useful in studies of electron-transfer reactions where solutions in a localized basis are preferred [43], The approach has recently been extended to also include matrix elements of a spin-orbit Hamiltonian. A number of RASSCF wave functions are used as a basis set to construct the spin-orbit Hamiltonian, which is then diagonalized [19, 44],... 

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

The energy expression (14) is a function of the molecular orbitals, appearing in the one- and two-electron integrals, and of the Cl expansion coefficients through the first- and second-order reduced density matrices (15). In an MCSCF optimization procedure the Cl coefficients and the parameters determining the MOs (normally the LCAO expansion coefficients) are varied until the energy reaches a stationary value. A number of procedures for performing the optimization have been described in the literature (see Ref. 26 for an extensive review). Here only the basic features of these procedures will be outlined, and the reader is referred to the literature for further details . ... 

In the second method, one expresses the matrix elements of the MC—SCF methods in terms of first-order and second-order transition density matrices connecting the N-electron basis functions matrix elements of the transition density matrices with respect to the n one-electron functions i, j, k, I forming the molecular orbitals used in the Cl expansion, are the coefficients of the one-electron and two-electron molecular integrals in the matrix element of the total Hamiltonian ... 

The exchange-correlation energy can thus be obtained by integrating the electron-electron interaction over the A variable and subtracting the Coulomb part. The right-hand side of eq. (B.18) can be written in terms of the second-order reduced density matrix eq. (6.14), and the definition of the exchange-correlation hole in eq. (6.21) allows the Coulomb energy to be separated out. 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

Given an N-electron wave function, the (S ) expectation value reads in terms of the (spinless) first- and second-order reduced density matrices, /)(rj rJ) and r, rp, respectively. 

He goes on to show that for the description of the energy it is sufficient to know the second-order reduced density matrix F(jc, jc2 i 2)- This was a favorite subject when he was lecturing and led to speculations about the possibility to compute the second-order density matrix directly, and discussions of the so called A/ -represent-ability problem. In spite of several attempts, this way of attacking the quantum chemical many-particle problem has so far been unsuccessful. Of special interest was the first-order reduced density matrix, 7( 1 jC ), which when expanded in a complete one-electron basis, ij/, is obtained as... 

In summary, this careful and systematic work has laid the groundwork for a topological force field (both inter and intra) drawn from ab initio electron densities and perhaps the second order reduced density matrix. Topological potentials are extractable and valid irrespective of computational... 

---