Biomedical subjects

Werner Kutzelnigg

Publications and source records attributed to Werner Kutzelnigg.

6 recordsLinked to original sources

PubMed2007

What I like about Hückel theory.

We start with some biographical notes on Erich Hückel, in the context of which we also mention the merits of Otto Schmidt, the inventor of the free-electron model. The basic assumptions behind the HMO (Hückel Molecular Orbital) model are discussed, and those aspects of this model are reviewed that make it still a powerful tool in Theoretical Chemistry. We ask whether HMO should be regarded as semiempirical or parameter-free. We present closed solutions for special classes of molecules, review the important concept of alternant hydrocarbons and point out how useful perturbation theory within the HMO model is. We then come to bond alternation and the question whether the pi or the sigma bonds are responsible for bond delocalization in benzene and related molecules. Möbius hydrocarbons and diamagnetic ring currents are other topics. We come to optimistic conclusions as to the further role of the HMO model, not as an approximation for the solution of the Schrödinger equation, but as a way towards the understanding of some aspects of the Chemical Bond.

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PubMed2006

Explicitly correlated potential energy surface of H3+, including relativistic and adiabatic corrections.

After a short historical account of the theory of the H3+ ion, two ab initio methods are reviewed that allow the computation of the ground-state potential energy surface (PES) of H3+ in the Born-Oppenheimer (BO) approximation, with microhartree or even sub-microhartree accuracy, namely the R12 method and the method of explicitly correlated Gaussians. The BO-PES is improved by the inclusion of relativistic effects and adiabatic corrections. It is discussed how non-adiabatic effects on rotation and vibration can be simulated by corrections to the moving nuclear masses. The importance of the appropriate analytic fit to the computed points of the PES for the subsequent computation of the rovibronic spectrum is addressed. Some recent extensions of the computed PES in the energy region above the barrier to linearity are reviewed. This involves a large set of input geometries and the correct treatment of the dissociation asymptotics, including the coupling with the first excited singlet state. Some comments on this state as well as on the lowest triplet state of H3+ are made. The paper ends with a few remarks on the ion H5+.

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PubMed2006

Density-cumulant functional theory.

Starting point is the energy expectation value as a functional of the one-particle density matrix gamma and the two-particle density cumulant lambda(2). We decompose gamma into a best idempotent approximation kappa and a correction tau, that is entirely expressible in terms of lambda(2). So we get the energy E as a functional of kappa and lambda(2), which can be varied independently. Approximate n-representability conditions, derived by perturbation theory are imposed on the variation of lambda(2). A nonlinear system of equations satisfied by lambda(2) is derived, the linearized version of which turns out to be equivalent to the coupled electron-pair approximation, variant zero. The start for kappa is Hartree-Fock, but kappa is then updated to become the best idempotent approximation of gamma. Relations to density matrix functional theory and Kohn-Sham type density functional theory are discussed.

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PubMed2005

Quasirelativistic theory equivalent to fully relativistic theory.

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to a quasirelativistic Hamiltonian matrix, that has the same electronic eigenstates as the original Dirac matrix. This transformation involves a matrix X, for which an exact identity is derived, and which can be constructed either in a noniterative way or by various iteration schemes, without requiring an expansion parameter. The convergence behavior of five different iteration schemes is studied numerically, with very promising results.

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PubMed2004

Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. III. Systems of noninteracting electrons.

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBCk) and the irreducible contracted Schrödinger equations (ICSEk) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC1 leads to a Liouville equation for the one-particle density matrix gamma1=gamma, consistent with our earlier analysis that the IBC1 holds both for a pure and an ensemble state. The IBC1 or the ICSE1 must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for gamma. For a closed-shell state gamma is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants lambdak with k> or =2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of lambda2, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of gamma. There are essentially three possibilities to construct the diagonal elements of gamma: (i) to consider the perturbation expansion of the characteristic polynomial of gamma, especially the idempotency for closed-shell states, (ii) to rely on the ICSE1, which (at variance with the IBC1) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.

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PubMed2004

Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. IV. Perturbative analysis.

The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schrödinger equations ICSEk for a closed-shell state are analyzed in terms of a Møller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC2(1), i.e., from the two-particle IBC to first order in the perturbation parameter mu, one gets the leading correction lambda2(1) to the two-particle cumulant lambda2 correctly. However, in order to construct the second-order energy E(2), one also needs the second-order diagonal correction gammaD(2) to the one-particle density matrix gamma. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE1(2); or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE3(2) to get lambda3(2), which is needed in the ICSE2(2) to get lambda2(2), which, in turn, is needed in the ICSE1(2) to get gamma(2). Generally the (k+1)-particle approximation is needed to obtain Ek correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring lambda(k+1) and lambda(k+2), by a renormalized hierarchy, in which only lambda(k+2) is ignored, and lambda(k+1) is expressed in terms of the lambdap of lower particle rank via the partial trace relation for lambda(k+2). Then the k-particle approximation is needed to obtain E(k) correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields E(k+1). We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of gamma(2) and two related matrices, in order to get estimates for gammaD(2) in terms of lambda2(1). In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of lambda2 in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E(2k-1). The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations.

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