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CRYSTAL is a periodic ab initio code that uses a Gaussian-type basis set to express crystalline orbitals (i.e., Bloch functions). The use of atom-centered basis functions allows treating 3D (crystals), 2D (slabs), 1D (polymers), and 0D (molecules) systems on the same grounds. In turn, all-electron calculations are inherently permitted along with pseudopotential strategies. A variety of density functionals are implemented, including global and range-separated hybrids of various natures and, as an extreme case, Hartree–Fock (HF). The cost for HF or hybrids is only about 3–5 times higher than when using the local density approximation or the generalized gradient approximation. Symmetry is fully exploited at all steps of the calculation. Many tools are available to modify the structure as given in input and simplify the construction of complicated objects, such as slabs, nanotubes, molecules, and clusters. Many tensorial properties can be evaluated by using a single input keyword: elastic, piezoelectric, photoelastic, dielectric, first and second hyperpolarizabilities, etc. The calculation of infrared and Raman spectra is available, and the intensities are computed analytically. Automated tools are available for the generation of the relevant configurations of solid solutions and/or disordered systems. Three versions of the code exist: serial, parallel, and massive-parallel. In the second one, the most relevant matrices are duplicated on each core, whereas in the third one, the Fock matrix is distributed for diagonalization. All the relevant vectors are dynamically allocated and deallocated after use, making the code very agile. CRYSTAL can be used efficiently on high performance computing machines up to thousands of cores. [ABSTRACT FROM AUTHOR]

In this paper we develop the shape effect, which is relevant for crystalline materials whose size is larger than that of the thermodynamic limit. According to this effect the electronic properties of one surface of a crystal depend upon all of its surfaces, i.e. on the overall shape. At first, qualitative mathematical arguments are presented for the existence of this effect based on the conditions for the stability of polar surfaces. Our treatment explains why such surfaces are observed even though earlier theory indicated that they should not exist. Then, models are developed from which it is found computationally that changing the shape of a polar crystal can substantially alter the magnitude of its surface charges. Apart from surface charges, it follows that the crystal shape will also significantly affect bulk properties, most notably polarization and piezoelectric responses. Additional model calculations show a strong shape effect on the activation energy for heterogeneous catalysis primarily through local surface charges rather than a non-local/long range electrostatic potential. [ABSTRACT FROM AUTHOR]

The response to electrostatic fields by finite systems, which contain a large central region of chemically identical building blocks, is shown not to depend on the boundaries of the system. Consequently, we can determine this response from that of the corresponding infinite periodic system and a vector potential theoretical-computational approach is presented for doing so. Both the electronic and structural response can be obtained with our approach. In many cases, neglecting the structural response is a poor approximation. To illustrate the performance of our treatment, extensive calculations have been carried out for simple model systems. Selected results are reported. Key words: polarization, extended systems, electrostatic fields, structural properties, electronic properties. On demontre que, pour des systemes finis contenant une grande region de blocs chimiquement identiques, la reponse a des champs electrostatiques ne depend pas des limites du systeme. En consequence, on peut determiner cette reponse a partir des systemes infinis correspondants et on presente une approche par calculs theoriques des potentiels vectoriels qui permet d'y arriver. Avec notre approche, il est possible d'obtenir aussi bien la reponse electronique que structurelle. Dans plusieurs cas, si on neglige la reponse structurelle, on obtient une mauvaise approximation. Afin d'illustrer la performance de notre traitement, on a effectue des calculs extensifs de systemes modeles simples. On rapporte des resultats choisis. Mots-cles : polarisation, systemes etendus, champs electrostatiques, proprietes structurelles, proprietes electroniques., [Traduit par la Redaction] Introduction With this contribution, we want to honor Tom Ziegler on the occasion of his 65th birthday. It happens that Tom Ziegler shares a number of [...]

Atheoretical treatment for the orbital response of an infinite, periodic system to a static, homogeneous, magnetic field is presented. It is assumed that the system of interest has an energy gap separating occupied and unoccupied orbitals and a zero Chern number. In contrast to earlier studies, we do not utilize a perturbation expansion, although we do assume the field is sufficiently weak that the occurrence of Landau levels can be ignored. The theory is developed by analyzing results for large, finite systems and also by comparing with the analogous treatment of an electrostatic field. The resulting many-electron Hamilton operator is forced to be hermitian, but hermiticity is not preserved, in general, for the subsequently derived single-particle operators that determine the electronic orbitals. However, we demonstrate that when focusing on the canonical solutions to the single-particle equations, hermiticity is preserved. The issue of gauge-origin dependence of approximate solutions is addressed. Our approach is compared with several previously proposed treatments, whereby limitations in some of the latter are identified. [ABSTRACT FROM AUTHOR]

For the computation of rovibrational levels and their spectroscopic intensities, the Eckart conditions are essential to achieve the optimal separation between rotation and vibration. Dymarsky and Kudin [J. Chem. Phys. 122, 124103 (2005)] proposed a procedure for a simplified calculation of the Eckart rotation matrix. In the present work, we have adapted their approach to obtain a kinetic energy operator in curvilinear coordinates using a numerical but exact procedure without resorting tofinite differences. Furthermore, we have modified this approach for the study of molecular systems with several minima, for which several Eckart reference geometries are required. The HONO molecular system has been used to show the efficiency of our implementation. Using the Eckart conditions with multi-reference geometries allows for a calculation of the rotational levels as well as frequencies and intensities of the infrared spectra of both HONO isomers with a single calculation. [ABSTRACT FROM AUTHOR]

We describe our implementation of a fully analytical scheme, based on the 2n + 1 rule, for computing the coupled perturbed Hartree Fock and Kohn-Sham dynamic first hyperpolarizability tensor β(-σ;ω1,ω2) of periodic 1D (polymer), 2D (slab), and 3D (crystal) systems in the CRYSTAL code [R. Dovesi et al., Int. J. Quantum Chem. 114, 1287 (2014)], which utilizes local Gaussian type basis sets. The dc-Pockels (dc-P) and second harmonic generation (SHG) tensors are included as special cases. It is verified that (i) symmetry requirements are satisfied; (ii) using LiF as an example, the infinite periodic polymer result agrees with extrapolated finite oligomer calculations and, likewise, for the build-up to a 2D slab and a 3D crystal; (iii) the values converge to the static case for low frequencies; and (iv) the Bishop-deKee dispersion formulas relating dc-P, SHG, and general processes are reproduced through quartic terms. Preliminary SHG calculations on multi-layer MoS2 satisfactorily reproduce experimental data. [ABSTRACT FROM AUTHOR]

The response to an electrostatic field is determined through simple model calculations, within both the restricted Hartree-Fock and density functional theory methods, for long, finite as well as infinite, periodic chains. The permanent dipole moment, μ0, the polarizability, α, and the hyperpolarizabilities β and γ , calculated using a finite-field approach, are extensively analyzed. Our simple model allows for treatment of large systems and for separation of the properties into atomic and unit-cell contributions. That part of the response properties attributable to the terminations of the finite system change into delocalized current contributions in the corresponding infinite periodic system. Special emphasis is placed on analyzing the reasons behind the dramatic overestimation of the response properties found with density functional theory methods presently in common use. [ABSTRACT FROM AUTHOR]

A fully analytical method for calculating Born charges and, hence, infrared intensities of periodic systems, is formulated and implemented in the CRYSTAL program, which uses a local Gaussian type basis set. Our efficient formalism combines integral gradients with first-order coupled perturbed Hartree-Fock/Kohn Sham electronic response to an electric field. It avoids numerical differentiation with respect to wave vectors, as in some Berry phase approaches, and with respect to atomic coordinates. No perturbation equations for the atomic displacements need to be solved. Several tests are carried out to verify numerical stability, consistency in one, two, and three dimensions, and applicability to large unit cells. Future extensions to piezoelectricity and Raman intensities are noted. [ABSTRACT FROM AUTHOR]

The large overshoot in (hyper)polarizabilities of quasilinear (1D) chains calculated by applying density functional theory with conventional functionals is investigated for several 2D and 3D extended systems. These systems include arrays of molecular hydrogen chains, as well as 2D coronene-type structures and LiF in 1D, 2D, and 3D. Contrary to a recently proposed model it is found that the overshoot persists in all of these cases. A simple explanation is provided by an analysis of the field-induced charges for molecular hydrogen, which shows an excessive buildup at the chain ends regardless of where the chain is located within the 2D and 3D array. [ABSTRACT FROM AUTHOR]

It is shown that the convergence of anharmonic infrared spectral intensities with respect to the basis set size is much enhanced in explicitly correlated calculations as compared to traditional configuration interaction type wave function expansion. Explicitly correlated coupled cluster (CC) calculations using Slater-type geminal correlation factor (CC-F12) yield well-converged dipole derivatives and vibrational intensities for hydrogen fluoride with basis set involving f functions on the heavy atom. Combination of CC-F12 with singles, doubles, and non-iterative triples (CCSD(T)-F12) with small corrections due to quadruple excitations, core-electron correlation, and relativistic effects yields vibrational line positions, dipole moments, and transition dipole matrix elements in good agreement with the best experimental values. [ABSTRACT FROM AUTHOR]

It is well known that the convergence of harmonic frequencies with respect to the basis set size in traditional correlated calculations is slow. We now report that the convergence of cubic and quartic force constants in traditional CCSD(T) calculations on H2O with Dunning’s cc-pVXZ family of basis sets is also frustratingly slow. As an alternative, we explore the performance of R12-based explicitly correlated methods at the CCSD(T) level. Excellent convergence of harmonic frequencies and cubic force constants is provided by these explicitly correlated methods with R12-suited basis irrespective of the used standard approximation and/or the correlation factor. The Slater type geminal, however, outperforms the linear r12 for quartic force constants and vibrational anharmonicity constants. The converged force constants from explicitly correlated CCSD(T) calculations succeed in reproducing the fundamental frequencies of water molecule with spectroscopic accuracy after corrections for post-CCSD(T) effects are made. [ABSTRACT FROM AUTHOR]

The high-frequency dielectric ε and the first nonlinear electric susceptibility χ(2) tensors of crystalline potassium dihydrogen phosphate (KH2PO4) are calculated by using the coupled perturbed Hartree–Fock and Kohn–Sham methods as implemented in the CRYSTAL code. The effect of basis sets of increasing size on ε and χ(2) is explored. Five different levels of theory, namely, local-density approximation, generalized gradient approximation (PBE), hybrids (B3LYP and PBE0), and HF are compared using the experimental and theoretical structures corresponding not only to the tetragonal geometry I4d2 at room temperature but also to the orthorhombic phase Fdd2 at low temperature. Comparison between the two phases and their optical behavior is made. The calculated results for the tetragonal phase are in good agreement with the experimental data. [ABSTRACT FROM AUTHOR]

The coupled perturbed Hartree–Fock (CPHF) method for evaluating static first (β) and second (γ) hyperpolarizability tensors of periodic systems has recently been implemented in the CRYSTAL code [Bishop et al., J. Chem. Phys. 114, 7633 (2001)]. We develop here an efficient and accurate computational protocol, along with the local basis sets needed for first and second row atoms. Application is made to several high symmetry three-dimensional systems including one (pyrope) with an 80 atom unit cell. CPHF second-order hyperpolarizabilities substantially undershoot experimental values, due to an overestimate of the band gap, but trends are satisfactorily reproduced for β as well as γ. [ABSTRACT FROM AUTHOR]

Various periodic piecewise linear potentials for extracting the electronic response of an infinite periodic system to a uniform electrostatic field are examined. It is shown that discontinuous potentials, such as the sawtooth, cannot be used for this purpose. Continuous triangular potentials can be successfully employed to determine both even- and odd-order (hyper)polarizabilities, as demonstrated here for the first time, although the permanent dipole moment of the corresponding long finite chain remains out of reach. Moreover, for typical highly polarizable organic systems, the size of the repeated unit has to be much larger than that of the finite system in order to obtain convergence with respect to system size. All results are illustrated both through extensive model calculations and through ab initio calculations on poly- and oligoacetylenes. [ABSTRACT FROM AUTHOR]

A general reduced dimensionality finite field nuclear relaxation method for calculating vibrational nonlinear optical properties of molecules with large contributions due to anharmonic motions is introduced. In an initial application to the umbrella (inversion) motion of NH3 it is found that difficulties associated with a conventional single well treatment are overcome and that the particular definition of the inversion coordinate is not important. Future applications are described. [ABSTRACT FROM AUTHOR]

A recently proposed approach for extracting band structures from finite-cluster calculations is improved so that (avoided) band crossings can be handled and the problems related to so-called doublings and holes are reduced. In particular, we demonstrate how the method can be combined with the elongation method for the finite-system calculations and apply it to extracting band structures for polymers from oligomer calculations. As illustrations of the approach we discuss a chain of water molecules, polyacetylene, polyethylene, and a BN nanotube without and with an impurity. [ABSTRACT FROM AUTHOR]

A computational scheme for the evaluation of the static first (β) and second (γ) hyperpolarizability tensors of systems periodic in 1D (polymers), 2D (slabs), 3D (crystals), and, as a limiting case, 0D (molecules) has been implemented, within the coupled perturbed Hartree–Fock framework (CPHF), in the CRYSTAL code, which uses a Gaussian type basis set. This generalizes to 2D and 3D the work by Bishop et al. (J. Chem. Phys. 114, 7633 (2001)). CPHF is applied for β and γ (the polarizability tensor α is also reported for completeness) of LiF in different aggregation states: finite and infinite chains, slabs, and cubic crystal. Correctness of the computational scheme and its numerical efficiency are documented by the trend of β and γ for increasing dimensionality: for a finite linear chain containing N LiF units, the hyperpolarizability tends to the infinite chain value at large N, N parallel chains give the slab value when N is sufficiently large, and N superimposed slabs tend to the bulk value. High numerical accuracy can be achieved at relatively low cost, with a dependence on the computational parameters similar to that observed for field-free self-consistent field (SCF) calculations. [ABSTRACT FROM AUTHOR]

A simple extended finite field nuclear relaxation procedure for calculating vibrational contributions to degenerate four-wave mixing (also known as the intensity-dependent refractive index) is presented. As a by-product one also obtains the static vibrationally averaged linear polarizability, as well as the first and second hyperpolarizability. The methodology is validated by illustrative calculations on the water molecule. Further possible extensions are suggested. [ABSTRACT FROM AUTHOR]

The long-range correction (LC) for treating electron exchange in density functional theory, combined with the Becke–Lee–Yang–Parr (BLYP) exchange-correlation functional, was used to determine (hyper)polarizabilities of polydiacetylene/polybutatriene oligomers. In comparison with coupled-cluster calculations including single and double excitations as well as a perturbative treatment of triple excitations, our values indicate that the tendency of conventional functionals to result in a catastrophic overshoot for these properties is alleviated but not eliminated. No clear-cut preference for LC-BLYP over Hartree–Fock values is obtained. This analysis is consistent with the calculations of Sekino et al. [J. Chem. Phys. 126, 014107 (2007)] on polyacetylene and molecular hydrogen oligomers. Thus, the performance of LC-BLYP with regard to (hyper)polarizabilities of quasilinear conjugated systems is now well characterized. [ABSTRACT FROM AUTHOR]

A method to build accurate band structures of polymers from oligomer calculations has been developed. This method relies on systematic procedures for (i) assigning k values, (2) eliminating strongly localized molecular orbitals, and (iii) connecting bands across the entire Brillouin zone. Illustrative calculations are carried out at the HF/STO-3G level for trans-polyacetylene (PA), poly(para-phenylene) (PPP), and water chains. More stringent tests at several different levels are reported for polydiacetylene/polybutatriene. [ABSTRACT FROM AUTHOR]

The vibrational configuration interaction method used to obtain static vibrational (hyper)polarizabilities is extended to dynamic nonlinear optical properties in the infinite optical frequency approximation. Illustrative calculations are carried out on H2O and NH3. The former molecule is weakly anharmonic while the latter contains a strongly anharmonic umbrella mode. The effect on vibrational (hyper)polarizabilities due to various truncations of the potential energy and property surfaces involved in the calculation are examined. [ABSTRACT FROM AUTHOR]

The response of periodic systems to external electric fields is a challenging theoretical problem. The authors show how the vector potential approach yields a numerically efficient treatment of the combined electronic and nuclear response to a finite static field. Their method is based on a self-consistent reformulation of the charge flow term in the single particle Hamiltonian. Careful numerical implementation yields a treatment whose computational needs are only marginally larger than those of a conventional field-free calculation. To prove the method exemplary polymer calculations are done for a model Hamiltonian. The latter contains all essential elements of an ab initio Kohn-Sham or Hartree-Fock Hamiltonian but allows for extensive testing. The extension to three-dimensional systems is described. [ABSTRACT FROM AUTHOR]

The static polarizability and second hyperpolarizability of increasingly large polydiacetylene and polybutatriene (PBT) chains have been evaluated using the optimized effective potential for exact exchange (OEP-EXX) method developed by Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)], where the unknown part of the effective potential is expressed as a linear combination of Gaussian functions. Various conventional atomic orbital basis sets were employed for the exchange potential (X basis) as well as for the Kohn-Sham orbitals [molecular orbital (MO) basis]. Our results were compared to coupled-perturbed Hartree-Fock (CPHF) calculations and to ab initio correlated values obtained at various levels of approximation. It turns out that (a) small conventional basis sets are, in general, unsatisfactory for the X basis; (b) the performance of a given X basis depends on the MO basis and is generally improved when using a larger MO basis; (c) these effects are exaggerated for the second hyperpolarizability compared to the polarizability; (d) except for the second hyperpolarizability of PBT chains, using 6-311++G** for the X basis gives reasonable agreement with the CPHF results for all MO basis sets; (e) our results suggest that in the limit of a complete X basis the OEP-EXX values may approach the CPHF data; and (f) in general, the quality of a given conventional X basis degrades with the length of the oligomer, which correlates with the fact that the number of X basis functions becomes a smaller fraction of the number required to reproduce exactly the finite-basis-set Hartree-Fock energies. Linear and especially nonlinear electric field responses constitute a very stringent test for assessing the quality of functionals and potentials; appropriately tailored basis sets are needed to describe the latter. Finally, this study further highlights the importance of electron correlation effects on linear and nonlinear responses, for which correlated functionals with OEP are required. [ABSTRACT FROM AUTHOR]

We have implemented our new procedure for computing Franck-Condon factors utilizing vibrational configuration interaction based on a vibrational self-consistent field reference. Both Duschinsky rotations and anharmonic three-mode coupling are taken into account. Simulations of the first ionization band of ClO2 and C4H4O (furan) using up to quadruple excitations in treating anharmonicity are reported and analyzed. A developer version of the MIDASCPP code was employed to obtain the required anharmonic vibrational integrals and transition frequencies. [ABSTRACT FROM AUTHOR]

A dipolar-free sum-over-states expression for the diagonal components of the first hyperpolarizability (β) tensor has been proposed by Kuzyk [Phys. Rev. A 72, 053819 (2005)] as an alternative to the traditional expression. We examine both alternatives for the longitudinal β of four typical push-pull π-conjugated systems using the ab initio CIS and CIS(D) schemes to approximate the excited state properties. Since they are each evaluated approximately the two SOS expressions yield different values for β and it is found that (i) they evolve symmetrically as the number of excited states is increased so that their average is nearly constant; (ii) in the static limit, the two values agree better with one another when their average is close to the “exact” correlated result; and (iii) frequency dispersion can affect the agreement between the alternative expressions. On the basis of (i) and (ii) it appears best for typical push-pull π-conjugated systems to estimate the static β, and the error in the value so obtained, by averaging the Kuzyk and traditional results. [ABSTRACT FROM AUTHOR]

Using a new unconventional procedure for calculating Franck-Condon factors with anharmonicity fully included the X 2A2←X 1A1 band in the photoelectron spectrum of furan (and deuterated furan) was simulated at the second-order perturbation theory level. All 21 vibrational modes were considered but, in the end, only 4 are required to accurately reproduce the spectrum. Except for our own recent work on ethylene such calculations have been previously limited to tri- or tetraatomic molecules. Most of the effect of anharmonicity is accounted for in first order, although second-order corrections to the vibrational frequencies are important. Based on these simulations we were able to improve upon and extend previous assignments as well as suggest further measurements. [ABSTRACT FROM AUTHOR]

The performance of the optimized effective potential procedure for exact exchange in calculating static electric-field response properties of push-pull π-conjugated systems has been studied, with an emphasis on NO2–(CH==CH)n–NH2 chains. Good agreement with Hartree–Fock dipole moments and (hyper)polarizabilities is obtained; particularly noteworthy is the chain length dependence for β/n. Thus, the problem that conventional density-functional theory functionals dramatically overestimate these properties is largely solved, although there remains a significant correlation contribution that cannot be accounted for with current correlation functionals. [ABSTRACT FROM AUTHOR]

A variational approach for reliably calculating vibrational linear and nonlinear optical properties of molecules with large electrical and/or mechanical anharmonicity is introduced. This approach utilizes a self-consistent solution of the vibrational Schrödinger equation for the complete field-dependent potential-energy surface and, then, adds higher-level vibrational correlation corrections as desired. An initial application is made to static properties for three molecules of widely varying anharmonicity using the lowest-level vibrational correlation treatment (i.e., vibrational Mo\ller–Plesset perturbation theory). Our results indicate when the conventional Bishop–Kirtman perturbation method can be expected to break down and when high-level vibrational correlation methods are likely to be required. Future improvements and extensions are discussed. [ABSTRACT FROM AUTHOR]

Our new simple method for calculating accurate Franck–Condon factors including nondiagonal (i.e., mode-mode) anharmonic coupling is used to simulate the C2H4+X 2B3u←C2H4X 1Ag band in the photoelectron spectrum. An improved vibrational basis set truncation algorithm, which permits very efficient computations, is employed. Because the torsional mode is highly anharmonic it is separated from the other modes and treated exactly. All other modes are treated through the second-order perturbation theory. The perturbation-theory corrections are significant and lead to a good agreement with experiment, although the separability assumption for torsion causes the C2D4 results to be not as good as those for C2H4. A variational formulation to overcome this circumstance, and deal with large anharmonicities in general, is suggested. [ABSTRACT FROM AUTHOR]

A different localization scheme for the elongation method is developed based on regional molecular orbitals. This scheme is more efficient and more accurate than the previous one especially for covalently bonded systems with strongly delocalized π electrons. Ab initio test calculations have been performed on three model systems: water chains, polyglycine, and cationic cyanine chains. The dependence on the size of the starting clusters and the effect of the basis set are investigated. Our results are compared with conventional ab initio calculations and it is found in all cases that the error per added unit levels off to a satisfactorily small value as long as the starting cluster is sufficiently large.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Using three typical π-conjugated molecules (1,3,5-hexatriene, 1-formyl-6-hydroxyhexa-1,3,5- triene, and 1,1-diamino-6,6-dinitrohexa-1,3,5-triene) we investigate the level of ab initio theory necessary to produce reliable values for linear and nonlinear optical properties, with emphasis on the vibrational contributions that are known to be important or potentially important. These calculations are made feasible by employing field-induced coordinates in combination with a finite field procedure. For many, but not all, purposes the MP2/6-31+G(d) level is adequate. Based on our results the convergence of the usual perturbation treatment for vibrational anharmonicity was examined. Although this treatment is initially convergent in most circumstances, a problematic situation has been identified. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An efficient new procedure for calculating Franck–Condon factors, based on the direct solution of an appropriate set of simultaneous equations, is presented. Both Duschinsky rotations and anharmonicity are included, the latter by means of second-order perturbation theory. The critical truncation of basis set is accomplished by a build-up procedure that simultaneously removes negligible vibrational states. A successful test is carried out on ClO[sub 2] for which there are experimental data and other theoretical calculations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A procedure to compute analytical second derivatives of the dynamic first hyperpolarizability with respect to vibrational coordinates has been developed and implemented at the time-dependent Hartree-Fock level. Simplifications are obtained by the same techniques as those employed to derive the 2n + 1 rule and the interchange relations. This scheme is used to determine the first-order ZPVA correction for three small molecules. It is found that the frequency dispersion coefficients are similar to those obtained for the pure electronic contribution in H[sub 2]P and NH[sub 3] but not CH[sub 4]. [ABSTRACT FROM AUTHOR]

We have studied how the calculation of electronic and vibrational contributions to nonlinear optical properties of three representative π-conjugated organic molecules is affected by the choice of basis set and the inclusion of electron correlation effects. The 6-31G basis does not always provide even qualitative accuracy. For semiquantitative accuracy a 6-31 + G(d) basis is sufficient. Although, as compared to QCISD, a second-order Møller-Plesset (MP2) treatment often yields a substantial fraction of the electron correlation contribution, our MP2 results for the separate electronic and vibrational terms are not consistently of semiquantitative accuracy. Nevertheless, at the MP2 level the ratio between the vibrational and electronic contributions is satisfactorily reproduced. [ABSTRACT FROM AUTHOR]

A treatment of nuclear vibrational motions in two-photon absorption is laid out in detail. Perturbation theory is used to develop working formulas for diatomic molecules and tested for the X[sup 1]&sigma[sup +]-A[sup 1]π transition of carbon monoxide. The results are compared with the exact numerical values. Certain vibrational contributions, not previously recognized, are found to make an important contribution to the two-photon absorption probability. [ABSTRACT FROM AUTHOR]