Extension of a simplified method for molecular correlation energy calculations to molecules containing third row atoms. I. Methodological developments

An extension of a simplified method for molecular correlation energy calculations to molecules containing third row atoms is presented. In addition to the use of pseudo-potentials in the calculations, the consequences of this extension on the different components of the energy partition which is the basic idea of the method, is analysed. Particular emphasis is placed on the specific role played by the 3d orbitals in each of the energy components. First, at the zeroth order, the energy is found to be very sensitive to the optimization of the 3d polarization functions. Secondly, the internal correlation energy, calculated by CI, requires the optimization of distinct 3d correlation orbitals to describe adequately the strong near-degeneracy effects that occur within the valence space. Finally it is shown that the 3d orbitals contribute partially to the non-internal correlation energy and that, the atoms-in-molecule structures corresponding typically to all-external contributions are negligible. The concept of error energy is introduced in place of the non-internal correlation energy: it includes the relativistic contributions within the semi-empirical tables. Such tables are presented for second row atoms and for the chlorine atom. From these tables, predicted values for some atomic term energies, experimentally undetermined, are derived. The methodological tests are limited here to the chlorine atom which is chosen for further applications in the next paper of this series. The conclusions concerning the applicability of the method to third row atoms are however quite general.Boursier I.R.S.I.A     

A method for molecular correlation energy calculations. Application to the determination of dissociation energies of diatomic and polyatomic molecules

A simple and economical method for molecular correlation energy calculations is developed. In this method, the internal part of the correlation energy is calculated by means of a CI in a minimal basis set and the non-internal part (semi-internal and all-external) is evaluated using an original atoms-in-molecule method. It is successfully applied to the determination of dissociation energies of some diatomic (H₂, NH, C₂, CN, N₂, CO, NO, O₂, F₂) and polyatomic (H₂O, N₂O, CO₂, N₃H, CH₂N₂, CH₂CO, C₂N₂) molecules. The results are compared to those obtained using very elaborate variational methods.Aspirant du Fonds National Belge de la Recherche Scientifique.     

Quantum-mechanical treatment of molecules. I. Calculations of the potential energy curves and molecular constants of the ground and ionized states of N2

The self-consistent-field molecular-orbital method in LCAO (linear combination of atomic orbitals) approximation is applied to the ground and three ionized states of N₂ at a number of internuclear distances for the computation of the potential energy curves. In these calculations both the linear coefficients and the screening constants of the atomic orbitals have been optimized. The molecular constants ω_e, ω_ex_e, B_e, α_e, and R_e have also been calculated for the above states from the computed potential energy curves. The computed spectral results are compared with the experimental data as well as with the results reported by others from ab initio calculations.     

Parametric method for calculating the vibronic spectra of complex molecules. Diphenylpolyenes

The absorption and fluorescence spectra of diphenylpolyenes (diphenylbutadiene, diphenylhexatriene, diphenyloctatetraene, and 1,6-di(4′-methylphenyl)-1,3,5-hexatriene) are calculated by a recently proposed parametric method using the fragmentation approach for designing molecular models. The parameters of the^H>C=molecular fragment (derivatives of the Coulomb and resonance integrals with respect to internal coordinates in the HAO basis set) obtained in calculation for polyenes were transferred to the molecular models of diphenylpolyenes without changes (∂H_e/∂q⁽⁰⁾=0.055 and ∂²H_e/∂q _k ⁽⁰⁾ ∂q _l ⁽⁰⁾ =0.1 au). The theoretical spectra are sufficiently adequate to quantitatively and qualitatively reproduce the main features of the vibrational structure of the experimental absorption and fluorescence spectra, and the parameters of the models of the potential surfaces of the excited states of diphenylpolyenes are consistent with the previous estimations. It is shown that this method allows predictive calculations of the vibronic spectra of complex molecules and the developed parametrization posesses all needed properties: locality, transferability, invariance to minor changes in electron density, ranking according to magnitude, small number of parameters for molecular fragments, etc. K. A. Timiryazev Moscow Agricultural Academy. Translated fromZhurnal Struktumoi Khimii, Vol. 37, No. 6, pp. 1040–1049, November–December, 1996. Translated by I. Izvekova     

Theoretical study of spectroscopic and molecular properties of several low‐lying electronic states of CO molecule

The potential energy curves (PECs) of eight low‐lying electronic states (X¹Σ⁺, a³Π, a′³Σ⁺, d³Δ, e³Σ^?, A¹Π, I¹Σ^?, and D¹Δ) of the carbon monoxide molecule have been studied by an ab initio quantum chemical method. The calculations have been performed using the complete active space self‐consistent field method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with the correlation‐consistent aug‐cc‐pV5Z basis set. The effects on the PECs by the core‐valence correlation and relativistic corrections are included. The way to consider the relativistic corrections is to use the third‐order Douglas–Kroll Hamiltonian approximation at the level of a cc‐pV5Z basis set. Core‐valence correlation corrections are performed using the cc‐pCVQZ basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are corrected for size‐extensivity errors by means of the Davidson modification (MRCI+Q). The spectroscopic parameters (D_e, T_e, R_e, ω_e, ω_ex_e, ω_ey_e, B_e, α_e, and γ_e) of these electronic states are calculated using these PECs. The spectroscopic parameters are compared with those reported in the literature. Using the Breit–Pauli operator, the spin–orbit coupling effect on the spectroscopic parameters is discussed for the a³Π electronic state. With the PECs obtained by the MRCI+Q/aug‐cc‐pV5Z+CV+DK calculations, the complete vibrational states of each electronic state have been determined. The vibrational manifolds have been calculated for each vibrational state of each electronic state. The vibrational level G(ν), inertial rotation constant B_ν, and centrifugal distortion constant D_ν of the first 20 vibrational states when the rotational quantum number J equals zero are reported and compared with the experimental data. Comparison with the measurements demonstrates that the present spectroscopic parameters and molecular constants determined by the MRCI+Q/aug‐cc‐pV5Z+CV+DK calculations are both reliable and accurate. © 2012 Wiley Periodicals, Inc.     

GRECP/MRD‐CI calculations of spin‐orbit splitting in ground state of Tl and of spectroscopic properties of TlH

The generalized relativistic effective core potential (GRECP) approach is employed in the framework of multireference single‐ and double‐excitation configuration interaction (MRD‐CI) method to calculate the spin‐orbit splitting in the ²P^o ground state of the Tl atom and spectroscopic constants for the 0⁺ ground state of TlH. The 21‐electron GRECP for Tl is used, and the outer core 5s and 5p pseudospinors are frozen with the help of the level shift technique. The spin‐orbit selection scheme with respect to relativistic multireference states and the corresponding code are developed and applied in the calculations. In this procedure both correlation and spin‐orbit interactions are taken into account. A [4,4,4,3,2] basis set is optimized for the Tl atom and employed in the TlH calculations. Very good agreement is found for the equilibrium distance, vibrational frequency, and dissociation energy of the TlH ground state (R_e=1.870 Å, ω_e=1420 cm⁻¹, D_e=2.049 eV) as compared with the experimental data (R_e=1.872 Å, ω_e=1391 cm⁻¹, D_e=2.06 eV). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 409–421, 2001     

A potential harmonic method for the three-body coulomb problem

A potential harmonic method that is suitable for the three-body coulomb systems is presented. This method is applied to solve the three-body Schroedinger equations for He and e ⁺ e ⁻ e ⁺ directly, and the calculations yield very good results for the energy. For example, we obtain a ground-state energy of −0.26181 hartrees for e ⁺ e ⁻ e ⁺, and −2.90300 hartrees for He with finite nuclear mass, in good agreement with the exact values of −0.26200 hartrees and −2.90330 hartrees. Compared with the full-set calculations, the errors in the total energy for ground and excited states of e ⁺ e ⁻ e ⁺ are very small, around −0.0001 hartrees. We conclude that the present method is one of the best PH methods for the three-body coulomb problem. Received: 5 September 1996 / Accepted: 14 July 1997     

Symmetry breaking and electron correlation in O2−, O2, and O2+: A comparison of coupled cluster and quadratic configuration interaction approaches

Potential energy curves are calculated for O₂⁻, O₂, and O₂⁺ at the CCSD, QCISD, CCSD(T), and QCISD(T) levels of theory using aug-cc-pVDZ and aug-cc-pVTZ basis sets with electron correlation built onto inversion symmetry constrained and relaxed UHF wave functions. The spectroscopic constant r_e, w_e, w_e, x_e, D_j, and α_e, are determined from the potential curves using standard second-order perturbation theory expressions and are compared with experimental values to assess the relative accuracy of the theoretical approaches. Comparison of corresponding symmetry-constrained and symmetry-relaxed calculations indicates that the CCSD method is generally superior to CCSD(T), QCISD, and QCISD(T) in recovering from a symmetry-broken reference function. © 1996 John Wiley & Sons, Inc.     

A systematic theoretical investigation of the valence excited states of the diatomic molecules B2, C2, N2 and O2

A quantitative survey on the performance of multireference (MR), configuration interaction with all singles and doubles (CISD), MRCISD with the Davidson correction and MR-average quadratic coupled cluster (AQCC) methods for a wide range of excited states of the diatomic molecules B₂, C₂, N₂ and O₂ is presented. The spectroscopic constants r _e, ω_e, T _e and D _e for a total of 60 states have been evaluated and critically compared with available experimental data. Basis set extrapolations and size-extensivity corrections are essential for highly accurate results: MR-AQCC mean-errors of 0.001 ?, 10 cm⁻¹, 300 cm⁻¹ and 300 cm⁻¹ have been obtained for r _e, ω_e, T _e and D _e, respectively. Owing to the very systematic behavior of the results depending on the basis set and the choice of method, shortcomings of the calculations, such as Rydberg state coupling or insufficient configuration spaces, can be identified independently of experimental data. On the other hand, significant discrepancies with experiment for states which indicate no shortcomings whatsoever in the theoretical treatment suggest the re-evaluation of experimental results. The broad variety of states included in our survey and the uniform quality of the results indicate that the observed systematics is a general feature of the methods and, hence, is molecule-independent. Received: 12 June 2000 / Accepted: 1 September 2000 / Published online: 21 December 2000     

Ab initio calculations of relativistic and electron correlation effects in polyatomics using the universal Gaussian basis set: XeF2

Ab initio accurate all-electron relativistic molecular orbital Dirac–Fock self-consistent field calculations are reported for the linear symmetric XeF₂ molecule at various internuclear distances with our recently developed relativistic universal Gaussian basis set. The nonrelativistic limit Hartree–Fock calculations were also performed for XeF₂ at various internuclear distances. The relativistic correction to the electronic energy of XeF₂ was calculated as ～ ?215 hartrees (?5850 eV) by using the Dirac–Fock method. The dominant magnetic part of the Breit interaction correction to the nonrelativistic interelectron Coulomb repulsion was included in our calculations by both the Dirac–Fock–Breit self-consistent field and perturbation methods. The calculated Breit correction is ～6.5 hartrees (177 eV) for XeF₂. The relativistic Dirac–Fock as well as the nonrelativistic HF wave functions predict XeF₂ to be unbound, due to neglect of electron correlation effects. These effects were incorporated for XeF₂ by using various ab initio post Hartree–Fock methods. The calculated dissociation energy obtained using the MP 2(full) method with our extensive basis set of 313 primitive Gaussians that included d and f polarization functions on Xe and F is 2.77 eV, whereas the experimental dissociation energy is 2.78 eV. The calculated correlation energy is ～ ?2 hartrees (?54 eV) at the predicted internuclear distance of 1.986 Å, which is in excellent agreement with the experimental Xe—F distance of 1.979 Å in XeF₂. In summary, electron correlation effects must be included in accurate ab initio calculations since it has been shown here that their inclusion is crucial for obtaining theoretical dissociation energy (D_e) close to experimental value for XeF₂. Furthermore, relativistic effects have been shown to make an extremely significant contribution to the total energy and orbital binding energies of XeF₂. © 1995 John Wiley & Sons, Inc.     

Construction of molecular orbitals localized on polyatomic fragments

A simple method of localizing molecular orbitals on polyatomic molecular fragments is proposed; the method allows one to separate orbitals in the structural units of extended molecules. The method is illustrated by semiempirical calculations of the binuclear bridged complexes [(NH₃)₅Rupy-(C₂H₂)_n-py-Ru(NH₃)₅]⁵⁺ (n = 0,1,2,3). One of possible application is construction of orbital bases for calculations by the configuration interaction method with limited sets of active MOs. Translated fromZhumal Strukturnoi Khimii, Vol. 39, No. 4, pp. 571–578, July–August, 1998.     

Theoretical electronic studies of aluminum monobromide and aluminum monoiodide

By using CASSCF/MRCI methods, theoretical molecular calculations have been performed for 12 electronic states for AlBr molecule and 12 electronic states for AlI molecule in the representation ^2s+1Λ (neglecting spin‐orbit effects). Calculated potential energy curves are displayed. Spectroscopic constants including the harmonic vibrational wave number ω_e, the electronic energy T_e referred to the ground state and the equilibrium internuclear distance R_e are predicted for these singlet and triplet electronic states for both AlBr and AlI molecules. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Core-valence correlation effects for molecules containing first-row atoms. Accurate results using effective core polarization potentials

The accuracy of employing effective core polarization potentials (CPPs) to account for the effects of core-valence correlation on the spectroscopic constants and dissociation energies of the molecules B₂, C₂, N₂, O₂, F₂, CO, CN, CH, HF, and C₂H₂ has been investigated by comparison to accurate all-electron benchmark calculations. The results obtained from the calculations employing CPPs were surprisingly accurate in every case studied, reducing the errors in the calculated valence D _e values from a maximum of nearly 2.5 kcal/mol to just 0.3 kcal/mol. The effects of enlarging the basis set and using higher-order valence electron correlation treatments were found to have only a small influence on the core-valence correlation effect predicted by the CPPs. Thus, to accurately recover the effects of intershell correlation, effective core polarization potentials such as the ones used in the present work provide an attractive alternative to carrying out computationally demanding calculations where the core electrons are explicitly included in the correlation treatment. Received: 11 May 1998 / Accepted: 27 July 1998 / Published online: 28 October 1998     

The externally contracted CI method applied to N2

An approximate multireference CI method is presented. By grouping together configurations with the same internal parts and freezing their relative weights by the use of perturbation theory, the number of variational parameters is drastically reduced. The loss of correlation energy is shown to be usually less than 2%, and the timing is less than one ordinary CI iteration. Examples from calculations on some states of the nitrogen atom and nitrogen molecule are given. The basis set convergence for the lowest excitation energy in the atom is very slow. Less than 50% of the correlation effect is obtained at the s, p, d limit. After the inclusion of ? functions this value is improved to 83%. The dissociation energies of the molecule also show slow basis set convergence with errors of 0.5 eV even after addition of ? functions. The bond distances are, howeever, accurately reproduced with errors of less than 0.005 Å for all the states. A qualitative discussion of predissociation in the a ¹Π_g and B ³Π_gstates caused by spin–orbit interaction with the ⁵Σ_g⁺ state, is finally presented. Rapidly oscillating lifetimes between the different vibrational states are predicted.     

A systematic preparation of new contracted Gaussian-type orbital sets. IV. The effect of additional 3s functions introduced by the use of the six-membered 3dGTOs

The Gaussian-type basis sets for molecular calculations are usually prepared by an atomic SCF program in which spherical coordinates are used. On the other hand, many molecular SCF and CI programs are written by using the Cartesian coordinates and as a result six-membered d-type functions (x², y², z², xy, yz, zx)e^?γr2 are often used. They contain one additional 3s function which does not exist in the atomic calculation. Therefore, we shall have an incorrect, deeper molecular binding energy, unless we readjust the atomic total energy by adding the 3s orbital to the original basis set. Some examples are shown in the case of Cu₂ molecule, where we have found that the correction is quite appreciable, which was overlooked in previous work.     

Accurate molecular electrostatic potentials based on modified PRDDO/M wave functions: III. Extension of the PESP method for calculation of electrostatic potential-derived atomic charges to compounds containing Li+, Na+, Mg2+, K+, Ca2+, Zn2+, and I

The PESP (Parameterized ElectroStatic Potential) method for calculating molecular electrostatic potentials, previously parameterized for H, C, N, O, F, P, S, Cl, and Br, is extended to molecules containing Li⁺, Na⁺, Mg²⁺, K⁺, Ca²⁺, Zn²⁺, and I. For a collection of 166 molecules containing 1668 atoms with at least one metal or iodine atom, PESP achieves an average absolute deviation in electrostatic potential-derived atomic charges of 0.042e⁻ compared with ab initio MP2/6-31G** calculations, with a correlation coefficient of 0.996. For a larger data set, consisting of 311 molecules encompassing all of the 16 elements just listed (2488 total atoms), PESP achieves an average absolute deviation of 0.040e⁻ and a correlation coefficient of 0.995. PESP calculations are an order of magnitude faster than the simplest ab initio method (STO-3G) on large molecules, while achieving a level of accuracy that rivals much more elaborate ab initio methods. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1456–1469, 1998     

Frequency and molecular-mass-dependent nonexponential proton NMR relaxation in polymer melts

A theoretical treatment of the nonexponential relaxation behavior of the different proton nuclear magnetic resonance (NMR) relaxation processes in polymer melts is presented. Formulas are derived for a three-component model given by two versions and a homogeneous distribution of correlation times. The theoretical results were tested with measurements of T₁, T_2e, and T₂ as functions of frequency and molecular mass in linear fractionated polyethylene samples. While the T₁ relaxation always yields exponential magnetization decays, the T_2e and T₂ measurements show biexponential relaxation behavior. From the calculations it was found that the correlation time of the local motion is independent of the molecular mass, whereas the correlation time of the slowest motional process increases with M^2.8_w for the three-component model and with M^2.2_w for the distribution of correlation times, respectively. © 1992 John Wiley & Sons, Inc.     

The ground-state potential curve for F2

The ground-state potential curve for F₂ has been obtained using large-scale MC SCF and CI methods. MC SCF curves were obtained with the CAS SCF method using a variety of sets of active orbitals. The main conclusion from the CAS SCF calculations is that the 2π_u orbital is important. CI curves were obtained using the contracted CI method. The largest calculations contained 312000 configurations proper spin and space (d_2h) symmetry. The main conclusions from the CI calculations are that the configuration XXX are important, otherwise errors in D_e of 0.3 eV and in r_e of 0.02 Å are found. The remaining errors at the CI level are 0.08 eV for D_e, 0.005 Å for r_e and less than 10 cm^?1 for the lowest vibrational levels.     

Dynamic correlation for MCSCF wave functions: An effective potential method

A method is suggested which allows the inclusion of dynamic correlation into CASSCF calculations. An effective Coulomb hole potential is added to the Hamiltonian. The potential has a simple form, which allows its implementation into existing LCAO programs using Gaussian integral packages. The parameters appearing in the potential are determined by fitting to empirical valence correlation energies for first row atoms. Calculations of ionization energies and electron affinities show considerable improvement compared to the MCSCF values. Test calculations on three molecules give the following results, H₂ r _e=0.745 (0.741) Å, D _e=4.62 (4.75) eV; N₂ r _e=1.099 (1.098) Å, D _e= 10.42 (9.91) eV; O₂ r _e=1.198 (1.207) Å, D _e=4.73 (5.21) eV. Experimental values within parenthesis. On leave from: Institute of Organic Chemistry, Polish Academy of Sciences, PL-01-224 Warszawa 42, ul. Kasprzaka 44, Poland.     

The calculation of electron localization and delocalization indices at the Hartree–Fock, density functional and post-Hartree–Fock levels of theory

 Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results. Received: 13 February 2002 / Accepted: 24 April 2002 / Published online: 18 June 2002