Mol. Phys. 81, 813-824 (1994)
(Received 5 July 1993; accepted 14 September 1993)
A new derivation is presented for the orientation-dependent interaction tensors which describe the electrostatic interaction between two multipole or multipole polarizability distributions both given in their own molecule-fixed coordinate system. This derivation leads to an explicit equation that allows an easy and direct evaluation of formulae for the components of the double-tensors also for higher ranks. Such an equation is given for the real, for the complex spherical and for the cartesian moments. The formula is suitable to provide stable computer-generated expressions for tensor operators of any rank.
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Chem. Phys. Lett. 219, 267-273 (1994)
(Received 17 December 1997)
Distributed atom-atom multipolar polarizabilities have been calculated at the coupled perturbed Hartree-Fock (CPHF) level, using Bader's topological theory of partition the molecular charge density into atomic domains. The proposed scheme applies without difficulty to molecules of arbitrary shape and symmetry and maintains a remarkable stability of the individual atomic polarizability components with respect to basis set extension, exemplified by the molecules CO, H2O, NH3 and BF3.
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Chem. Phys. Lett. 233, 359-370 (1995)
(Received 17 November 1994)
A new method for the calculation of correlated frequency-dependent polarizabilities by the use of time-dependent second-order Møller-Plesset perturbation theory is presented. Within this method the dynamic polarizabilities are defined as second derivatives of the MP2 energy with respect to the strengths of two oscillating external fields. This definition assures the symmetry of mixed polarizabilities and leads to the same second-order pole structure as the diagrammatic double-perturbation MBPT methods. As derivative method this approach becomes equivalent to finite field MP2 calculations in the static limit. We report first applications to helium and neon.
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J. Phys. Chem. 100, 6243-6348 (1996)
Received: September 20, 1995; In Final Form: January 12, 1996
Using the recently developed time-dependent second-order Møller-Plesset perturbation theory (TDMP2), we have calculated correlated frequency-dependent multipole polarizabilities for the noble gases Argon through Radon. To get accurate results, i.e., near the TDMP2 basis set limit, we considered large one particle basis sets, optimized for polarizability calculations. With the exception of Argon, where we assumed relativistic effects to be neglegible, we employed the spin-free Hamiltonian obtained by means of a Douglas-Kroll transformation to treat relativistic and correlation effects on equal footing. Where reference data is available, the results for the static values and for the first cauchy moments of all polarizabilities considered are in good agreement with the best theoretical and experimental data. Utilizing these results, we have calculated improved dispersion coefficients for interactions involving the rare gas atoms Ar through Rn, which are estimated to be accurate within a few percent.
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Mol. Phys. 88, 69-92 (1996)
(Received 23 October 1995; accepted 30 December 1995)
Certain difficulties with the usual one-centre multipole expansion of long-range intermolecular interaction energies can be circumvented by multi-centre multipole expansions using several expansion sites in each molecule, such as the nuclear positions for example. Based on the topological partitioning of the molecular volume provided by Bader's `Atoms in Molecules' theory we have recently developed a method to calculate the required atomic multipole moments and polarizabilities. We study the performance of these toplogically partitioned electric properties for the calculation of multipole expanded first-order electrostatic and second-order induction energies by comparing their convergence behaviour to that of the corresponding one-centre expansions. The homomolecular dimers of the water, carbon monoxide, cyanogen, and urea molecules serve as examples. The results show that distributed electric properties calculated within the topological partitioning scheme indeed solve the `shape' convergence problem, which arises in the calculation of interaction energies of large non-spherical molecules via multipole expansions.
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Can. J. Chem. 74, 976-987 (1996)
Received October 13 1995.
Frequency-dependent distributed polarizabilities have been determined from time-dependent Hartree-Fock calculations, using the partitioning of the molecular space suggested by Bader's topological theory of atoms in molecules. The basis set dependence of the distributed dynamic polarizabilities is analyzed in terms of the first few Cauchy moments, for the carbon monoxide, water, cyanogen, urea and benzene molecules. Two alternative relocalization schemes have been considered in order to reduce the number of distributed dynamic polarizability parameters. The first one, closely related to the atomic polarizability model of Bader, leads to atomic charge-dipole and dipole-dipole polarizabilities, describing the response of the molecular charge distribution to a uniform external field, in terms of atomic charges and dipoles. The second scheme, similar to that suggested by Stone, retains the fully distributed description of the dynamic charge-flow polarizabilities, while all two-center dipole-dipole and charge-dipole contributions are condensed in one-center dynamic dipole-dipole polarizabilities.
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Mol. Phys. 89, 139-156 (1996)
(Received 2 January 1997; accepted 8 February 1996)
Relativistic and non-relativistic all-electron, ab initio methods are used to investigate the role of the individual energy components in the total interaction energy involved in the formation of the weakly bound ground state X 1Sigma+g of the Hg2 molecule. The interaction energy is partitioned according to a hybrid approach into a supermolecular repulsive potential and the damped dispersion energy derived from perturbation theory. Both parts are then computed individually and their dependence on relativity and electron correlation is investigated. From this analysis, hybrid potentials that comprise different physical interactions are constructed and the importance of specific features is evaluated by comparison of the appropriate hybrid potentials with each other and with the experimental curve. The most detailed model is based on a CASSCF supermolecular potential to which the damped intramonomer correlated dispersion energy series is added. In the relativistic case it yields a potential with quantitatively correct asymptotic behaviour and which is able to predict potential parameters that agree reasonably well with the experimental results. The asymptotic density method developed by Duman and Smirnow was tested as an inexpensive alternative for the calculation of the repulsive part of the hybrid potential. In spite of its recent success in the calculation of the He-He potential, it cannot provide a good repulsive component in the case of Hg2. As a result of this work, this failure can attributed to the unusually strong induction effects in the ground state mercury dimer bond, which lower repulsive forces substantially.
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Mol. Phys. 89, 595-605 (1996)
(Received 1 December 1995; accepted 17 March 1996)
Distributed polarizabilities of a series of n-alkanes CnH2n+2 (n = 2-7) in various conformations have been determined using Bader's topological theory of atoms in molecules. Using an appropriate localization scheme, a simples distributed model is constructed, where the methyl and methylene groups are characterized by their dipole polarizability tensors in local frames, and all charge flow polarizabilities between them are retained. A set of average polarizability parameters is proposed that takes into account the local environment of the methyl and methylene groups, and that not only reproduces the polarizability tensor for any member of the n-alkane series in an arbitrary conformation, but also is suitable for the calculation of induction energies.
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J. Chem. Phys. 105, 9948-9965 (1996)
(Received 30 April 1996; accepted 26 August 1996)
In order to test the performance of the recently developed time-dependent second-order Møller-Plesset perturbation theory (TDMP2) for anisotropic frequency-dependent multipole polarizabilities, we have studied the isoelectronic series CO, N2, CN- and NO+. The polarizabilities of these triplebonded diatomics are an interesting test for the TDMP2 method, because it is known that Møller-Plesset perturbation theory has its difficulties to describe multiple bonds. We selected these molecules as test systems, because especially for N2 and CO accurate experimental and other ab initio data is available to compare with and their dynamical polarizabilities are needed for dispersion coefficients of van der Waals complexes, which are presently under intensive investigation. To get reliable results near the TDMP2 basis set limit we used large one-particle basis sets, optimized for polarizability calculations on the coupled Hartree-Fock level. The results show that the TDMP2 method is capable to improve for the isotropic as well as for the anisotropic polarizabilities considerable on the TDHF approximation, with the exception of the dipole polarizabilities of N2 and NO+, for which the static correlation effects are too strong to be treated by second-order Møller-Plesset perturbation theory. However, we find, that the TDMP2 method due to the use of coupled (TDHF) first-order orbital rotation parameters is somewhat more stable with respect to static correlation effects than the SDT-MBPT(2) double perturbation theory. Where reference data is available, the TDMP2 results for static polarizabilities and for the first Cauchy moments are in good agreement with the best theoretical and experimental data. We also calculated dispersion coefficients for the (N2)2 and the (CO)2 dimer and, utilizing the results of previous TDMP2 studies for the atoms He through Xe, also for the respective rare gas complexes of N2, CO, CN- and NO+. We estimate the results to be the most accurate ab initio data available for these van der Waals coefficients.
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Chem. Phys. Lett. 260, 341-351 (1997)
(Received 11 July 1996; in final form 12 August 1996)
Recurrence relations for the orientation-dependent multipole interaction tensors are derived in the spherical double-tensor formalism. These relations make it possible to calculate all interaction double-tensor components through order L in an expansion in 1/R with computational expenditures that scale as L4. In contrast to the cartesian interaction tensors the orientation-dependent spherical double-tensors make it possible without any further manipulation to evaluate multipole expanded electrostatic and induction energies with algorithms that also scale as L4. By introducing an intermediate transformation to a special coordinate system the matrix of spherical interaction tensor elements can be factorized into a product of three sparse matrices, each of which can be calculated within L3 steps. Employing this factorization, we devise algorithms for the calculation of electrostatic, induction and dispersion energies that scale through order $L$ in the multipole expansion as L3, L4 and L5, respectively.
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Mol. Phys. 91, 145-160 (1997)
Multicentre multipole expansions allow to solve the
`shape' convergence problem arising in the calculation of long-range
interaction energies between large non-spherical molecules via point-multipole
expansions. In paper I of this series
(1996, Molec. Phys., 88 69) it has been shown that
this is the case for first-order electrostatic and second-order induction
energies when employing distributed multipole moments and static
polarizabilities generated from topological partitioning of the molecular
volume as provided by Bader's `atoms-in-molecules' theory.
Their generalization to frequency-dependent, topologically partitioned
polarizabilities is used in the present contribution to compare the
convergence behaviour of one-centre and multicentre multipole expansions
of the second-order dispersion energy for homonuclear dimers of the water,
carbon monoxide, cyanogen and urea molecules. The findings are similar to
those for the induction energy: the radial `extension' convergence
problem, which exists already for point-multipole expanded interaction energies
between atoms, necessarily persists, but the angular convergence problems
linked to the shape of interacting molecules can succesfully be treated
by multicentre multipole expansions of the dispersion energy.
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Chem. Phys. Lett. 268, 521-530 (1997)
(Received 13 January 1997; in final form 18 February 1997)
We derive efficient algorithms for the derivatives, in particular the forces, torques and the hessian, for coulomb-type interactions and for general anisotropic pair potentials. We use spherical tensor techniques and an intermediate transformation to a quasi-internal coordinate system, proposed recently to obtain a factorization of the multipole interaction tensors. By these means we obtain a unified approach to the derivatives at small extra computational cost to the energy calculation. For the special case of an electrostatic interaction the computational cost scales for a system of N molecules through rank $L$ in the multipole expansion as N2 × O(L3) for both the energy and its derivatives.
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Chem. Phys. Lett. 269, 428-434 (1997)
(Received 10 December 1996; in final form 14 March 1997)
The first implementation of frequency-dependent hyperpolarizabilities is presented using coupled cluster quadratic response theory. The quadratic response functions are derived for the coupled cluster model hierachy CCS, CC2 and CCSD and implemented for general frequency-dependent first hyperpolarizabilities. Calculations are performed on the second harmonic generation and the optical rectification hyperpolarizabilities of FH and the results are compared with experimental results and former theoretical investigations.
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J. Chem. Phys. 107, 10592-10598 (1997)
(Received 13 May 1997; accepted 19 September 1997)
Expressions for the even Cauchy moments for nonvariational methods have been derived using the time-averaged quasienergy Lagrangian technique. The expressions obtained require the solution of linear equations but do not involve a sum over individual excited-state contributions. An implementation is reported for the coupled cluster models CCS, CC2, and CCSD and calculations have been performed for the Cauchy moments and the Verdet and Cotton-Mouton constants of the Ne atom and for the C6 dispersion coefficient of the Ne2 dimer.
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Chem. Phys. Lett. 282, 139-146 (1998)
(Received 27 June 1997; revised 22 October 1997; accepted 24 October 1997)
The coupled cluster cubic response function is derived using a time-averaged quasienergy Lagrangian and Fourier component variational perturbation theory. The obtained expressions are in accord with the 2n+1 and 2n+2 rules. The first implementation of the cubic response function is reported for frequency-dependent second hyperpolarizabilities for a coupled cluster singles and doubles wavefunction and sample calculations are presented for hyperpolarizabilities of N2 for several third-order optical processes. The results are in excellent agreement with recent experiments for electric field induced second harmonic generation.
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Chem. Phys. Lett. 107, 109-113 (1998)
(Received 13 October 1997; revised 4 November; accepted 14 November 1997)
The frequency-dependent electric field induced second harmonic generation hyperpolarizability of Neon is calculated using coupled cluster singles doubles (CCSD) response theory. The CCSD dispersion curve shows that the most recent fit through the experimental measured ESHG data is too flat due to the fact that one of the measured points in the infrared region is incorrect. Based on the CCSD dispersion curve, we suggest an adjusted extrapolation of the experimental ESHG data to a static limit result of 106.4±2 a.u. This estimate is corroborated by a CCSD(T) calculation using a large one-particle basis set and including core correlation effects that gives 107.1 a.u.
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submitted for publication to J. Chem. Phys.
Based on an analysis of the first residues of coupled cluster response functions we devise variational functionals from which the transition moments for n-photon excitations can be calculated as n-th. derivatives. Combining these functionals with variational perturbation theory, we obtain a new approach for the derivation of multiphoton transition moments which allows to utilize the full strengths of variational perturbation theory without the roundabout way via residues of response functions. Coupled cluster multiphoton transition moments derived by this approach are formally equivalent to the one identified from the first residues of the ground state response functions. The introduction of the variational functionals makes the mathematical structure of the transition moments more transparent and provides an interpretation of intermediates in terms of responses of excited state vectors and Lagrangian multipliers. 2n+1 and 2n+2 rules are formulated for the transition moments and build the basis for a straightforward derivation of a computational efficient formulation. The strength of the new approach is demonstrated by the derivation of three- and fourphoton transition moments.
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submitted for publication to J. Chem. Phys.
The implementation of twophoton transition moments is described for the hierachy of coupled cluster models CCS, CC2 and CCSD and applications are reported for the twophoton transition probability rate constants for transitions from the ground state to the first excited 1Se state in helium and to the lowest 1De and 1Se states in neon and argon. At the CCSD level the rate constant for the 1De transition in argon is in good agreement with the experimental rate constant and also with a recent MCSCF calculation.
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Mol. Phys., in press
(Received 12 December 1997; accepted 22 January 1998)
The frequency-dependence of experimentally-derived or point-wise ab initio calculated hyperpolarizability averages is often expanded up to fourth order in the frequencies using the dispersion formula X||(n)(w1;...,wn) = X||(n)(0) (1 + A wL2 + B wL4 + O(wi6) ) with wL2 = sumi wi2, where the coefficient A is independent of the optical process. We derive an extension of this dispersion formula which is open ended in the powers of the frequencies, uses only process-independent coefficients, and involves a minimal number of terms per order. It is shown that the dispersion formula can be cast into a form where dispersion coefficients drop out in a systematic manner for optical processes involving static electric fields. We discuss how the concept of these dispersion formulas can be generalized to other hyperpolarizability components and exemplify this approach at the experimentally important components beta\bot and betaK of the first hyperpolarizability.
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submitted for publication to J. Chem. Phys.
The recently developed coupled cluster cubic response approach is applied to the calculation of frequency-dependent second hyperpolarizabilities of Ne, Ar, N2 and CH4. A basis set investigation is performed for the static limits of the hyperpolarizabilities at the SCF and the coupled cluster singles doubles level using series of correlation consistent basis sets. The role of correlation effects on the dispersion of the hyperpolarizabilities is explored by a comparison of SCF, CCS, CC2 and CCSD results for third harmonic generation. Dispersion curves are calculated at the CCSD level for the Kerr effect, degenerate four wave mixing, electric field induced second harmonic generation, and third harmonic generation. To get a realistic comparison with experiment we have taken into account zero point vibrational averaging, pure vibrational contributions and corrections for core correlation and connected triples. For all four examples we obtain much better agreement with experiment than in previous ab initio calculation.
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