The one-electron density matrix

The densities (1.7.5) constitute the elements of an Af x A/ Hermitian matrix - the one-electron spin-ori ital density matrix D - since the following relation is satisfied  

The one-electron density matrix is positive semidefinite since its elements are either trivially equal to zero or inner products of states in the subspace F(M, N — 1). The diagonal elements of the spin-orbital density matrix are the expectation values of the occupation-number operators (1.3.1) in F(M, N) and are referred to as the occupation numbers Top of the electronic state  

This terminology is appropriate since the diagonal elements of D reduce to the usual occupation numbers kp in (1.3.2) whenever the reference state is an eigenfunction of the ON operators - that is, when the reference state is an ON vector  

Since the ON operators are projectors (1.3.5), we may write the occupation numbers in the form 

Recalling that the occupation numbers kp of an ON vector are zero or one, we conclude that the occupation numbers cop of an electronic state (1.7.13) are real numbers between zero and one - zero for spin orbitals that are unoccupied in all ON vectors, one for spin orbitals that are occupied in all ON vectors, and nonintegral for spin orbitals that are occupied in some but not all ON vectors  

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We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

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. 

For this limited Cl problem, it is convenient to diagonalize the one-electron density matrix, defining a set of natural orbitals that vary at each step of an iterative procedure. The one-electron density operator is defined by... 

The resonance contribution to the energy of each bond is proportional to the off-diagonal element of the one-electron density matrix known as the Coulson bond order between the HOs of the m-th geminal J eq. (2.78) ... 

The last row is most important here. It demonstrates that in the linear response approximation the off-diagonal matrix element of the one-electron density matrix (the Coulson bond order) does not change, (i.e. is invariant even for the different atoms forming the bond and even more to the geometry changes). This result suggests the stability of the bond orders with some precision. However, this result should... 

In the QM part of the system, the variation of the bond orders can also take place. In variance with the pure SLG picture [11,12] used here as the QM method underlying the MM part of the system, the atoms in the QM part of the combined system may have off-diagonal elements of the one-electron density matrix between orbitals ascribed to the QM subsystem. The latter are obviously the (Coulson) bond orders for the QM part of the system. The corresponding contribution to the energy reads ... 

The one-electron density matrix corresponding to the solution of the Hartree-Fock problem in the CLS is, like any Hartree-Fock density matrix, an operator (matrix) P... 

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