Geometrical energy derivative evaluation with MRCI wave functions

The theory of MCSCF and CI energy derivatives with respect to geometrical variations is briefly reviewed with special attention given to the MCSCF and MRCI energy gradients. A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF ” equations, it does not require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not require any derivative integrals to be transformed from the AO basis to the MO basis. An additional feature is that it does not require any changes to existing MCSCF gradient evaluation programs in order to compute MRCI gradients. The only difference in the two cases is the exact nature of the data passed to the gradient evaluation program from the previous steps in the computational procedure. The additional effort required to compute the entire MRCI energy gradient vector is approximately that required for one additional iteration of the MRCI diagonalization procedure and for one additional MCSCF iteration. For large scale MRCI wave functions, the MRCI energy gradient evaluation should only require about 10% of the effort of computing the wave function itself. This computational procedure removes a major computational botleneck of potential energy surface evaluation.     

Generalization of analytic energy derivatives for configuration interaction wave functions

This paper takes the form of a review including some original contributions. A fresh derivation of analytic energy derivative expressions for configuration interaction (CI) wave functions is presented. In this method the CI energy is described by _IJC_IC_J(H _IJ-_IJE) so that the orthonormality condition is explicitly included therein. In the sequence of differentiations up to fourth order it will be demonstrated that each derivative may be expressed in terms of (H _IJ-_IJE) and its derivatives in a symmetric way with respect to the interchange of differential variables. In a similar manner, the CI variational condition may be described in an equation which explicitly includes the normalization condition. It is shown that the differentiation of the modified variational condition produces the coupled perturbed configuration interaction (CPCI) equations in directly soluble and compact forms. The necessary formulae for the energy derivatives up to fourth order and the CPCI equations up to second order are explicitly given.     

Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.     

Correlation energy extrapolation by intrinsic scaling. III. Compact wave functions

The information gained in the context of extrapolating the correlation energy by intrinsic scaling is used to shorten the full configurational expansions of electronic wave function without compromising their chemical accuracy. The truncations are accomplished by judiciously limiting the participation of the ranges of predetermined approximate sets of natural orbitals in the various excitation categories.     

Efficient calculation of potential energy surfaces for the generation of vibrational wave functions

An automatic procedure for the generation of potential energy surfaces based on high level ab initio calculations is described. It allows us to determine the vibrational wave functions for molecules of up to ten atoms. Speedups in computer time of about four orders of magnitude in comparison to standard implementations were achieved. Effects due to introduced approximations--within the computation of the potential--on fundamental modes obtained from vibrational self-consistent field and vibrational configuration interaction calculations are discussed. Benchmark calculations are provided for formaldehyde and 1,2,5-oxadiazole (furazan).     

Vibrational partition functions for H2O derived from perturbation-theory energy levels

Purely vibrational energy levels and partition functions are calculated using three different potential energy surfaces for the H₂O molecule. Results obtained with perturbation-theory, independent-normal-mode (INM), and harmonic approximations are compared with accurate values. For the cases considered here, the expected improvement that perturbation theory provides over the corresponding harmonic treatment is found to be substantial, while the INM approximation leads to results which are worse than the corresponding harmonic ones. In fact, we show that reliable partition functions for these potential surfaces can be obtained when resonance contributions are removed from the perturbation-theory treatment, and we propose a theoretical criterion for deciding when a particular interaction should be treated as resonant.     

Optimal Hylleraas wave functions

Hylleraas wave functions composed of the optimally combinedN terms (2 N 20) are presented for two-electron atoms with nuclear chargesZ = 1 (H^–), 2(He), 3(Li⁺), 5(B³⁺), and 10(Ne⁸⁺). The spherically-averaged electron density (r) and electron-pair densityh(r ₁₂) are constructed in a simple and analytical functional form from the 20-term functions. Comparison of several one- and two-electron moments r ^k and r ₁₂ ^k shows that the present density functions have near-exact accuracy.     

Two by two rotations of orbitals for minimizing the energy of LCAO-MO wave functions

A new method is studied for finding the molecular orbitals which minimize the energy of an LCAO-MO wavefunction. The method makes use of successive rotations of pairs of orbitals. It can be applied to multi-determinant as well as to single-determinant wavefunctions. Criteria are found to minimize excited states.

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Zusammenfassung Es wird eine neue Methode zur Bestimmung der M.O.s nach dem Kriterium minimaler Gesamtenergie beschrieben. Dabei werden jeweils zwei M.O.s sukzessive transformiert. Das Verfahren bleibt auch dann anwendbar, wenn für die Zustandsfunktion eine Linearkombination von HS-Determinanten angesetzt wird. Zum Schluß werden noch einige Kriterien für die Bestimmung von angeregten Zuständen angegeben.

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Résumé Une nouvelle méthode est étudiée ici pour trouver les orbitales moléculaires qui minimisent l'énergie d'une fonction d'onde LCAO-MO. La méthode utilise des rotations successives de couples d'orbitales. La méthode peut minimiser aussi des fonctions d'onde multi-déter-minantales et les états excités.

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The author is indebted with Professor B. Pullman and Dr. A. Pullman for their suggestions and helpful advises. He aknowledges also Dr. Berthier for useful discussions.

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This work prepared in part at the Institut de Biologie Physico-Chimique, Université de Paris, was supported by a grant CY-3073 of the US Public Health Service (National Cancer Institute) to that Institute.     

Excited state wave functions

Wave functions and energies were calculated for the 2s, 3p ₀, and 4d ₀ states of the hydrogen atom using the Messmer and Rayleigh-Ritz variational methods with minimization of the second eigenvalue. The wave functions were linear expansions of Gaussian functions and both linear and exponential parameters were varied. Except for the two term expansions, calculated values of the energies and expectation values, r ^–1, r and r ² were within two percent of the true values for both methods.     

Laplace transform wave functions

The wave function defining a quantum-mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen-like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.     

Local energy as an indicator of the accuracy of wave functions and probability densities for helium

The local energy is examined as an indicator of the accuracy of approximate wave functions for the ground state of helium. It is observed that at a given point (1) an inaccurate local energy may or may not correspond to an inaccurate value of the wave function or probability density, but (2) a value of the local energy within 0.1 a.u. of the ground-state energy corresponds to a value of the approximate wave function or probability density within about 10% of that for the ground-state wave function.     

Systematic errors in the total energy of molecular wave functions calculated within the PRDDO approximations

We compare calculated total energies for 150 open-chain molecules using ab initio methodology and the PRDDO approximations. The bulk of the errors implicit in the PRDDO approximations are apparently of a one-center nature, i.e., they are due to the number and type of atoms in the molecule, and not the details of the molecular geometry. Atomic correction factors are developed which reduce the errors in the calculated total energy of PRDDO wave functions by a factor of eight relative to the ab initio reference calculations. PRDDO calculations on ring and cage compounds are shown to have additional systematic errors in the total energy.     

Analytic expression of the second derivatives of electronic energy for full configuration interaction wave functions

Summary A new analytic second derivative expression of the electronic energy is derived for full configuration interaction (CI) wave functions. This formula is shown to be free from the derivative terms of both CI and MO coefficients. The second-order relationships between CI and MO coefficients for full CI wave functions are also presented.     

Linear integrability of wave functions

Finiteness is usually imposed as a condition for physical admissibility of a wave function. Examination of this condition in both position- and momentum-space shows that finiteness of the wave function at the origin in one space implies “linear” integrability in the reciprocal space, except in some pathological cases. Some implications of this result are discussed.     

Electronic wave functions for atoms

Several specialized configuration interaction (CI) calculations for the ground states of the He isoelectronic series have been carried out with the purpose of defining successive orders of approximation to the wave function, so that reliable patterns of convergence can be investigated for the energy, some one-electron expectation values, and the wave function itself. We advocate the use of a sequence of wave functions to extrapolate expectation values and to find the extrapolation error = final error bound. As a direct consequence of this study, we show what the utmost limitations of CI expansions are for these systems and what is to be expected in similar situations (electron pairs in many-electron wave functions). Finally, a comparison is made between CI and interparticle coordinates wave functions.

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Zusammenfassung Für die isoelektronische Reihe des He werden eine Reihe spezieller CI-Rechnungen für die Grundzustände ausgeführt mit dem Ziel, eine Folge von Näherungen an die Wellenfunktion definieren zu können. Auf diese Weise können zuverlässige Kriterien für das Konvergenzverhalten für die Energie, einige Ein-Elektronen-Erwartungswerte und für die Wellenfunktion untersucht werden. Wir befürworten den Gebrauch einer Folge von Wellenfunktionen, um die Erwartungswerte extrapolieren zu können und den Fehler der Extrapolation (= die endgültige Fehlergrenze) zu finden. Darausfolgend zeigen wir für diese Systeme die Grenzen von CI-Reihen und was in ähnlichen Fällen (Elektronenpaaren in Mehr-Elektronen-Wellen-Funktionen) zu erwarten ist. CI Funktionen und Funktionen, die die Relativkoordinaten der Teilchen enthalten, werden verglichen.

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Résumé On a effectué plusieurs calculs d'interaction de configuration spécialisés pour les états fondamentaux de la série isoélectronique à He, en vue de déterminer la fonction d'onde avec différents degrés d'approximation pour étudier avec sûreté la convergence de l'énergie, de certains observables monoélectroniques et de la fonction d'onde elle même. Nous avons recours à une suite de fonctions d'onde pour extrapoler les valeurs moyennes et trouver la limite d'erreur. Cette étude a pour conséquence directe de montrer que les limitations les plus sévères du traitement d'I.C. apparaissent pour ces systèmes et sans doute pour ceux présentant une situation analogue (paires électroniques dans des fonctions d'onde polyélectroniques). Finallement, on compare l'I.C. et les fonctions d'onde contenant les coordonnées interparticulaires.

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This work was supported by grants from the National Science Foundation and the US Air Force Office of Scientific Research.     

Hypervirial theorems and restrictions on wave functions

The conditions that ensure that an optimal variational wave function ø under general restricting requirements satisfies the hypervirial theorem are analysed. Application is made to a system where thez -component of angular momentum is a constant of motion and results are discussed in connection with those obtained via symmetry considerations.     

Electronic computation of first-order wave functions using green's functions

Green's functions and the symbol manipulative computer language LISP have been used to obtain exact, closed form, first-order functions and second-order energies for the first fourteen states of the hydrogen atom in electric and magnetic fields.     

Torsional potential and strain energy distribution in an extended chain under stress

Torsional potentialV() for the single bond transformation in an extended hexadecane, subjected to elongation, has been determined by molecular mechanics calculations. The stored elastic energy significantly modifies the potentialV(), the conformational energies and the barriers of transition. Apart from the soft torsional coordinate, elastic energy is also dissipated considerably by bond stretching and angle bending. Maximal variations of the valence coordinates occur in the vicinity of the torsional defect and dampen along the chain. At higher elongation, the gauche minimum on the potentialV() disappears and the calculations predict the abrupt gauche to trans transition. The energetics of torsion of a deformed chain are compared with the experimental data on the hydrodynamic extension of polymers in dilute solution by elongational flow. The calculations also provide details of a single bond transformation mechanism at conformational interconversions in a long chain, proposed by Helfand.