Hartree-Fock natural orbitals

The natural orbital 0i is equivalent to the variational Hartree-Fock Is orbital in this case, much closer to the exact hydrogenic solution discussed in Section 1.2. 

The electronic structures of poiy(fluoroacetylene) and poly(difluoroacetylene) have been investigated previously using the ab initio Hartree-Fock crystal orbital method with a minimum basis set (42). Only the cis and trans isomers with assumed, planar geometries were studied. The trans isomer was calculated to be more stable in both cases, and the trans compounds were predicted to be better intrinsic semiconductors and more conductive upon reductive doping than trans polyacetylene. However, our results show that head-to-tail poly(fluoroacetylene) prefers the cis structure and that the trans structure for poly(difluoroacetylene) will not be stable. Thus the conclusions reached previously need to be re-evaluated based on our new structural information. Furthermore, as noted above, addition of electrons to these polymers may lead to structural deformations that could significantly change the conductive nature of the materials. 

Exercise 4.8 For the special case of a two-electron system, the use of natural orbitals dramatically reduces the size of the full Cl expansion. If is the occupied Hartree-Fock spatial orbital and r = 2,3,..., K are virtual spatial orbitals, the normalized full Cl singlet wave function has the form... 

As mentioned in Sect. 2, the exchange-overlap energy depends on the nature of the spin coupling of the interacting molecules [18, 19]. For closed-shell molecules the resultant total spin is zero, and the first-order contribution to the exchange-overlap component of the interaction can be expressed in closed form if Atp is approximated as a single determinant of Hartree-Fock spin-orbitals of the individual molecules [48-50]. 

My personal special emphasis has always been on the wavefunction itself. Since the wavefunction is not an observable, it is not possible to carry out an empirical calibration of a model wavefunction. Rather one must place it in the context of a sequence of wavefunctions that ultimately converges to the exact answer and produces correct properties without empirical corrections. At the same time, I prefer wavefunctions that apply to as wide a range of molecular systems as possible but that have some chance of being interpreted. The Cl wavefunctions generated for small molecules using natural or MCSCF orbitals are of this type. More modern wavefunctions such as MPn, full Cl, or coupled clusters calculated with Hartree-Fock virtual orbitals are not interpretable, and are usually never even looked at. 

Almost all contemporary electronic structure calculations involve the use of one-electron functions called spin orbitals. The calculations usually commence with the computation of canonical Hartree-Fock (spin) orbitals ( molecular orbitals or MOs) which are eigenfunctions of the Fock operator. In turn, MOs are input into complicated functionals that yield approximate values of the correlation energy. Formal differentiation of those functionals with respect to strengths of external perturbations results in formulae for effective one-electron density matrices F(r, r ) 3 -t33 from which any one-electron, first-order response property can be calculated. Finally, diagonalization of F(r, i ) produces natural (spin) orbitals (NOs) (see Natural Orbitals). 

The natural orbital concept, as originally formulated by Per-Olov Lowdin, refers to a mathematical algorithm by which bestpossible orbitals (optimal in a certain maximum-density sense) are determined from the system wavefunction itself, with no auxiliary as sumptions or input. Such orbitals inherently provide the most compact and efficient numerical description of the many-electron molecular wavefunction, but they harbor a type of residual multicenter indeterminacy (akin to that of Hartree-Fock molecular orbitals) that somewhat detracts from their chemical usefulness. 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

Hartree-Fock (momentum space) 139 natural orbitals 27... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

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