Paul Westhaus

First principles derivation of the effective valence shell Hamiltonian for large molecules

We construct an effective Hamiltonian for the valence shell of a large electronic system. The procedure begins by classifying the complete one electron space according to core, valence, and excited orbitals. An N−electron subspace TN spanned by Slater determinants with a fixed set of Nc core orbitals and different sets of Nv = (N − Nc) valence orbitals is defined. A canonical transformation on the Coulomb Hamiltonian is used to eliminate the interaction between TN and its orthogonal complement, ?N, thereby defining an effective N−electron Hamiltonian. This effective Hamiltonian is expanded in a cluster development of linked one−, two−, three−,⋅⋅⋅body operators in terms of which the conditions of having vanishing matrix elements between TN and ?N can be explicitly formulated. Then starting with this effective N−electron Hamiltonian we construct an equivalent Hamiltonian to operate in the space of antisymmetrized products of valence orbitals. To within a constant (times the identity on the valence space) th...

Effective valence shell interactions in carbon, nitrogen, and oxygen atoms

We present the first numerical results of the canonical tranformation formalism for generating the valence shell effective interactions. Considering the valence space to be formed as the antisymmetrized direct product of 2s and 2p type orbitals we have computed all the matrix elements of the two‐body valence shell interactions for isolated carbon, nitrogen, and oxygen atoms. Explicit forms for the core, valence, and excited orbitals are the hydrogenlike functions defined by an effective nuclear charge. We compare our results with the semiempirical ’’electron–electron repulsion parameters’’ finding that for reasonable values of the effective charge—essentially a choice of the core and valence spaces—good agreement with the parameters deduced from atomic spectra data is obtained. Not only do the general features of semiempirical theory such as the reduction of the average strength of the two‐body interaction appear, but also the somewhat more subtle effects such as the relative ordering of the diagonal matr...

Excitation spectra of atoms and small molecules using effective valence shell Hamiltonians generated by canonical transformations

The canonical transformation–cluster expansion formalism is used to generate the effective valence shell Hamiltonian for carbon. Hydrogenlike orbitals defined by an effective nuclear charge parameter Z are used to span the core (K shell), valence (L shell), and excited (3⩽n⩽9) spaces. The effective Hamiltonian containing one‐ and two‐body interactions is diagonalized on the Nv‐particle valence space to yield the low‐lying excitation spectrum. Considering alternative approximations to carry out the calculations, we indicate the importance of including the two‐body pair potential as well as the single particle operators in the generator of the canonical transformation. Upon doing this, good agreement with experiment is obtained for the lowest valence shell transitions over a wide range of Z. For certain physically reasonable Z the entire valence shell experimental excitation spectrum can be accurately reproduced. In contrast, a ’’zeroth order’’ effective Hamiltonian using only the ’’charge cloud’’ of the co...

An effective Hamiltonian for the valence states of ethylene generated by canonical transformations

We calculate the vertical N→T, N→V, and T→V transition energies in the ethylene molecule by diagonalizing the effective valence shell Hamiltonian obtained by our first principles canonical transformation–cluster expansion formalism. Calculations are performed for five different partitions of the one‐electron space into core, valence, and excited subspaces. Excellent results of 4.61, 7.67, and 3.06 eV are obtained for the three respective transitions, provided that the many‐electron states arising from a given partition meet two criteria. In particular, we find it is necessary to include both a localized and a diffuse π* orbital explicitly in the valence space. In addition, care must be taken in partitioning the one‐electron space so as to avoid the quasidegeneracy between a state in the model space and one of the same symmetry in the orthogonal complement.

Semiempirical effective Hamiltonians from a first principles’ viewpoint

We extend the formal apparatus for generating effective electronic Hamiltonians by a unitary (canonical) transformation on the Coulomb Hamiltonian to model spaces having various numbers of valence electrons. This formal structure is then used to analyze the problem of deducing effective Hamiltonians from empirical data. It is found that certain ambiguities generally arise since there is no way of empirically establishing the unitary transformation within the model space. At best, one can find a family of unitarily equivalent effective Hamiltonians. Examples are given for the π‐electron system of ethylene, and the implications of this point of view are discussed for understanding semiempirical theories of electronic molecular structure.

The constrained variational formulation of energy functionals

We present a generalization of the density functional formalism based upon constrained variations of the expectation value E=〈Ψ‖H‖Ψ〉. The essential idea is to find the minimum value of E with respect to a set of state vectors Sa={‖Ψ〉a}, each member of which yields the specified expectation values ai≡a〈Ψ‖Ai‖Ψ〉a of set of observables Ai, i=1...M. a denotes the collection of M expectation values ai, i=1...M. If one or more N‐electron vectors exist yielding the given M expectation values, a is said to be N representable. One thus obtains this minimum expectation value E(a)=min E over Sa as a function of the assigned expectation values. In terms of the function E(a), the exact ground state energy may be obtained by seeking its minimum value over all possible (N‐representable) expectation values a. The traditional density functional formalism of Hohenberg and Kohn is seen to be a special case where the constraints are expressed by the expectation values of an indenumerable set of charge density operators, the...

Hydrogen bonds and masking in the sulfhydryl group in proteins

Abstract Szent-Gyorgi has shown that the relative amounts of readily titratable (“free”) SH groups vs. those readily titratable only following denaturation (“masked”) varies significantly from normal to cancerous organ tissues. It is therefore important to inquire into the nature of the two forms of protein-borne SH. Of the four suggested mechanisms for the “masking” of protein SH groups toward hydrophilic reagents, namely: (i) sequestration in hydrophobic regions—whether between chain-folds or between agglomerated protein sub-units, (ii) local steric hindrance, (iii) cyclic hydrogen-bonding to local peptide amino acid residues, or (iv) covalent bonding as in thiazoldines; the first mechanism, that of sequestration in hydrophobic regions appears from present evidence to be the most likely cause. Various spectroscopic, reaction rate and entropy arguments are presented and compared which lead to this conclusion. In addition we have calculated the binding energy of the SH group to various sets of lone pair electrons appropriate to N, O, F, P, S, and Cl atoms in molecules. The calculation was made in the configuration interaction of valence orbitals (CIVO) scheme and gave binding energies and interatomic distances in reasonable agreement with available experimental data.

Constrained energy functionals obtained as solutions to a partial differential equation

The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖H0‖Ψ〉a}, ai=a〈Ψ‖Ai‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to‐be‐found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrodinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.

Normal modes of a two-dimensional lattice of interacting dipoles

AbstractWe derive the normal mode frequency spectra