Matrix-Based Energy Functional

As discussed, for a formulation of SCF theories suitable for large molecules, it is necessary to avoid the nonlocal MO coefficient matrix, which is conventionally obtained by diagonalizing the Fock matrix. Instead we employ the one-particle density matrix throughout. For achieving such a reformulation of SCF theory in a density matrix-based way, we can start by looking at SCF theory from a slightly different viewpoint. To solve the SCF problem, we need to minimize the energy functional of 

These conditions are automatically fulfilled upon diagonalization of the Fock or the Kohn-Sham matrices and the formation of the density matrix (Eq. [8]). 

The question now becomes How do we impose these properties without diagonalization Li, Nunes, and Vanderbilt (abbreviated as LNV) first realized in the context of tight-binding (TB) calculations (see also the related work of Daw ) that insertion of a purification transformation introduced by McWeeny in 1959120,121 one to incorporate the idempotency con- 

In the derivation of density matrix-based SCF theory below, we do not employ the chemical potential introduced by LNV, but instead we follow the derivation of Ochsenfeld and Head-Gordon, because McWeeny s purification automatically preserves the electron number.Therefore, to avoid the diagonalization within the SCF procedure, we minimize the energy functional 

At convergence this energy gradient expression reduces to the usual criterion of FPS — SPF = 0. It is important to note that the covariant energy gradient (Eq. [117]) cannot be added directly to the contravariant one-particle density matrix, so that a transformation with the metric is required. 

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A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

Compared to computational approaches based on the g-electron density, approaches based on the g-electron reduced density matrix have the advantage that the kinetic energy functional can be written in an explicit form ... 

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

The effect of temperature on the bulk structure can be studied by free energy calculations and by crystal dynamics simulations. Infra-red and Raman spectra, and certain inelastic neutron scattering spectra directly reflect aspects of the lattice dsmamics. Infra-red spectra can be simulated firom the force constant matrix, based on interatomic potential models [94-97]. The matching of simulated mode fiequencies with those measured in Raman or IR spectra can indeed be used to develop, validate or improve the form and parameterization of the interatomic potential functions [97]. 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

If one can model the noninteracting one-matrix yj(r,r ) as a functional of the electron density, then, using Equation 1.47 or Equation 1.48, one can compute the kinetic energy. This is the most straightforward approach to deriving kinetic energy functionals, and the Thomas-Eermi functional and the Weizsacker functional can both be derived in this way. (Indeed, all of the most popular functionals can be derived in several different ways.) The one-matrix can also be modeled based on weighted density approximation (WDA), which we will discuss subsequently. 

This assumption is based on a more fundamental assumption eoneeming the elastie energy (stored energy function). If the elastic energy, which is a potential function for the stress tensor, vanishes in the unstrained state and can be expressed by a symmetric quadratic form, then the stiflness matrix is symmetric, i.e. the elasticity tensor is fully symmetric. 

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