An efficient approach for the calculation of Franck-Condon integrals of large molecules

A general and efficient approach for the calculation of Franck-Condon integrals (FCIs) of large molecules is presented. In a first step, by exploiting the diagonally dominant and sparse structure of the Duschinsky matrix, a model system is constructed for which the Duschinsky matrix takes a block-diagonal form. For each of these blocks separately, the FCIs are calculated discarding all below a certain threshold. From those integrals retained the FCIs of the model system are obtained by simple multiplication. These serve as an estimate for the FCIs of the exact system which are calculated for those integrals which lie above a certain threshold. By systematically decreasing the threshold, the simulation can be reliably converged to the exact result with an arbitrary accuracy. Using this scheme, a considerable reduction of the number of FCIs which have to be calculated is achieved which leads to an improved scaling behavior of the computational effort with system size. The approach has been tested thoroughly for a set of molecules including difficult cases. For the larger systems a speedup of up to three orders of magnitude compared to an exact calculation is observed while the errors can be kept negligible. With this approach accurate calculations of FCIs are feasible also for large molecules encountered in "real-life" chemistry, especially biochemistry and material science.     

Recursion formulae for calculation of overlap integrals

Multi-ζ Slater-type orbitals are frequently used in molecular orbital calculations. Master formulae and numerical tables are available in literature for overlap integrals between s, p, and d atomic orbitals up to principal quantum number (n) = 3 and for some other selected quantum numbers. However, no master formula or numerical table is available for quantum numbers n = 5 and above and involving ? orbitals. In this article recursion formulae have been presented for the calculation of the overlap integral between any two s, p, d, and ? atomic orbitals formed by a linear combination of Slater-type orbitals. These formulae, when expanded, would give rise to all the master formulae reported in the literature as well as formulae hitherto unreported.     

The calculation of Franck-Condon factors for the electronic transitions of the polyacenes

Progressional intensities have been calculated for the ¹ L _a band of the polyacenes which give good agreement with experiment. The Franck-Condon factors are calculated on the assumption that the frequencies and normal modes in the ground and excited states are identical. Our results, based on a simple force field, also show that the assumption (by McCoy and Ross) that the C-C stretching frequencies are degenerate is satisfactory as far as the calculation of the shape of the overall vibrational envelope is concerned.

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Zusammenfassung Wir berechneten die Intensitäten der ¹ L _a-Progressionen von Acenen, in guter Übereinstimmung mit dem Experiment. Die Franck-Condon-Faktoren werden unter der Annahme bestimmt, daß Frequenzen und Normalschwingungen im Grund- und angeregten Zustand gleich sind. Unsere mit einem einfachen Kraftansatz erhaltenen Ergebnisse zeigen weiter, daß die Annahme (von McCoy und Ross) entarteter C-C-Valenzschwingungen ausreicht, um die allgemeine Schwingungsstruktur der Bandenenveloppe zu berechnen.

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Résumé Nous avons calculé les intensités de progression de la bande ¹ L _a des acènes, en bon accord avec l'expérience. Les facteurs de Franck-Condon sont déterminés, en admettant des fréquences et vibrations normales identiques pour les états fondamental et excité. Nos résultats, sur base d'un champ de force simple, montrent de plus que l'hypothèse (de McCoy et Ross) des vibrations de valence C-C dégénérées est satisfaisant pour le calcul de la forme générale de l'enveloppe vibrationelle des bandes.

     

Asymptotic calculation of some exchange integrals

Asymptotic calculations of the exchange integrals with excitation transfer and without excitation transfer are carried out. The atoms are considered in the ground or excited states. The effect of the spin of the atomic core is taken into account.

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Zusammenfassung Es werden asymptotische Ausdrücke für Austauschintegrale berechnet, wobei auch Übergang von Anregung eingeschlossen wird. Ferner wird der Einfluß der Spins der Atomrümpfe behandelt.

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Résumé Calcul asymptotique des intégrales d'échange avec et sans transfert d'excitation. Les atomes sont envisagés dans l'état fondamental ou les états excités. L'effet du spin sur le coeur atomique est pris en considération.

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The autors are indebted to Dr. E. E. Nikitin and Dr. M. Ja. Ovchinnikova for very valuable discussions.     

Statistical computer-aided calculation of molecular integrals

An improved Monte Carlo procedure is described for the evaluation of molecular integrals, which is particularly suitable for multicenter and/or two-electron calculations. The method is almost independent of the complexity of the atomic orbitals involved, and the results can be obtained with an uncertainty which is fairly adequate for most applications and with very moderate waste of computer resources. In the version presented here there is only a restriction concerned with the positive value of some functions involved, as described in the text, but this possibility does not arise in much of the practical work. An example with a two-dimensional exchange integral is worked out in detail.     

Modified rotation transformation method for the calculation of repulsion integrals

A modified rotation transformation method for two electron repulsion integrals is introduced in this paper. The percentage of nonzero matrix needed to perform the calculation is less than 30 in this method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Modified rotation transformation method for the calculation of repulsion integrals

A modified rotation transformation method for two electron repulsion integrals is introduced in this paper. The percentage of nonzero matrix needed to perform the calculation is less than 30 in this method.     

Perturbative calculation of intermolecular interactions in orthogonalized or biorthogonal basis sets

Two versions of a many-body perturbation theory for the computation of molecular interaction energies are investigated. The methods are based on the partitioning of the second-quantized form of the dimer Hamiltonian written either in the orthogonalized basis of the monomer MOs, or, alternatively, in the original non-orthogonal dimer basis set handling the overlap by the biorthogonal formalism. The zeroth-order Hamiltonian H ⁰ is the sum of effective monomer Fockians and the zeroth-order wave functions are exact eigenfunctions of H ⁰. Full antisymmetry is ensured by the second-quantized formalism. Basis set superposition error is accounted for by the counterpoise correction recipe. Results for He₂, (H₂)₂ and (H₂O)₂ indicate the reliability of the biorthogonal technique.     

Improved algorithm for the calculation of one-electron two-center integrals with STOs

In a previous article (J. Fernández Rico, R. López and G. Ramírez, J. Comp. Chem., 9 , 790 (1988)) we have proposed the calculation of molecular integrals involving STOs by means of some recurrence relations which use two sets ( h and H ) of overlap integrals (basic matrices). In the present paper, we derive explicit expressions of these integrals employing the two-range expansion of the 0s-function. This approach yields equations for the elements of the two basic matrices in terms of two further matrices, k (x,y) and i (x,y), and some auxiliary functions. Relations between the elements of these matrices and the functions are thoroughly explored and numerical tests are included for illustrating the behavior of the method.     

Molecular integrals in the approximate calculation of electronic structure

A self-consistent set of new proposals is made for the calculation of the largest molecular integrals over orthogonal hybrid orbitals used in neglect of differential overlap schemes.     

Franck-Condon simulations of clusters: phenol-nitrogen

Multidimensional Franck-Condon simulations of the resonance enhanced multiphoton ionization (REMPI) and mass-analyzed threshold ionization (MATI) spectra of phenol-nitrogen are obtained from CASSCF, MRCI, and SACCI optimized geometries. In the REMPI simulations, the results are unsatisfactory, as the transitions associated with intermolecular modes are widely underestimated and much less intense than those associated with intramolecular modes. Conversely, the simulations of the MATI spectra show a good similarity to experiment. The best simulations are obtained in both instances from the SACCI optimized geometries. Furthermore, the simulations suggest that the two most prominent Franck-Condon envelopes present in the MATI spectra are due to the sigma and sigma + ngamma' combination bands in accord with the assignments of the MATI spectra of the analogous phenol-carbon monoxide cluster.     

Perturbative calculation of the Hartree–Fock interaction energy using orthogonalized orbitals

A many‐body perturbation theory based on the partitioning of the dimer Hamiltonian, formulated in an orthogonalized basis set, is used for the calculation of interaction energies at the Hartree–Fock (HF) level. Numerical results for the (HF)₂ and (H₂O)₂ systems in selected geometries are presented. The interaction‐energy components are compared with the results obtained from the standard supermolecular approach and the intermolecular perturbation theory based on the biorthogonal basis set. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 81–88, 1999