Self-consistent Field Values Research Articles

Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: Exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.

Interaction Electric Hyperpolarizability Effects in Weakly Bound H2O···Rg (Rg = He, Ne, Ar, Kr and Xe) Complexes†

We report an extensive high-level ab initio theoretical investigation of the interaction electric properties of the weakly bound complexes of water with rare gases, H(2)O...Rg (Rg = He, Ne, Ar, Kr and Xe). Our approach relies on finite-field many-body perturbation theory (MP) and coupled-cluster (CC) calculations with flexible, carefully designed basis sets of Gaussian-type functions (GTFs). We have obtained estimates of the electron correlation effects on the reference self-consistent field (SCF) values at all levels of theory employed in this work. For the planar equilibrium configuration (C(s) symmetry) of the complexes, at the CCSD(T) level of theory, the interaction mean first hyperpolarizability is positive (SCF values in parentheses): beta(int)/e(3)a(0)(3)E(h)(-2)(H(2)O-Rg) = 0.72 (0.56), 1.20 (0.79), 7.04 (4.28), 9.88 (5.34), and 15.52 (6.68) for Rg = He, Ne, Ar, Kr, and Xe, respectively. For the interaction mean second hyperpolarizability, we have obtained (SCF values in parentheses): gamma(int)/e(4)a(0)(4)E(h)(-3)(H(2)O-Rg) = -18.93 (-14.19), -37.78 (-24.35), -83.16 (-72.42), -102.92 (-125.70), and -206.45 (-286.83) for Rg = He, Ne, Ar, Kr, and Xe. In addition to the equilibrium values of the interaction electric properties, we have also extracted information for their dependence on the variation of the distance of the rare gas atom from the center of mass of the water molecule (without changing direction). The gradient of the interaction mean second hyperpolarizability at the equilibrium configuration is determined as (dgamma(int)/dR)(e)/e(4)a(0)(3)E(h)(-3)(H(2)O-Rg) = 9.89 (He), 25.42 (Ne), 73.71 (Ar), 144.71 (Kr), and 324.68 (Xe).

On the electric properties of Br2

Electric multipole moments and polarizabilities for Br2 are obtained from finite field self consistent field (SCF), many-body perturbation theory and coupled cluster calculations with large gaussian-type basis sets. At the MP2 level of theory the bond length dependence of the mean and the anisotropy of the dipole polarizability (ααβ/e 2 a 0 2 E h -1) around the equilibrium bond length of Re = 2·28105 A is adequately described by , . A very large \\[18s15p12d6f] basis set consisting of 330 basis functions yields SCF values α = 44·10 and Δ α = 28·31 e 2 a 0 2 E h -1. Adding electron correlation corrections obtained at the CCSD(T) level of theory and vibrational corrections obtained at the MP2 level, the mean dipole polarizability at the ground vibrational state is estimated at α = 44·74 (to be compared with the experimental value of 45·0) and the anisotropy at Δ α = 25·39 e 2 a 0 2 E h -1. At the MP2 level the mean second dipole hyperpolarizability γ/ e 4 a 0 4 E h -1 varies around R e as . SCF, MP2 and CCSD(T) results are combined to give as final theroeticalprediction the estimate of γ = (10·2 ± 1·0) x 103 e 4 a 0 4 E h -3.

Accurate electric multipole moments for HCN and HCP from CCSD(T) calculations with large Gaussian basis sets

Accurate values of the electric multipole moments of HCN and HCP have been obtained from self-consistent field (SCF) and coupled-cluster (CCSD(T)) calculations. With the origin at the centre of mass and hydrogen along the positive molecular axis in both systems, a [9s5p2d/10s7p5d3f/10s7p5d3f] basis set is expected to predict near-Hartree-Fock values for the dipole (μ=1.2962ea0), quadrupole (Θ=2.1046ea 0 2 ), octopole (Ω=10.088ea 0 3 ) and the hexadecapole (Φ=24.23ea 0 4 ) moment of HCN. An analogous basis set, [9s5p2d/10s7p5d3f/14s11p7d3f], predicts SCF values of μ=0.1421ea0, Θ=3.8786ea 0 2 , Ω=19.633ea 0 3 and Φ=65.89ea 0 4 for HCP. Electron correlation reduces the dipole moment of HCN but increases the dipole moment of HCP. At the CCSD(T) level of theory the calculated values are μ=1.1800ea0, Θ=1.6461ea 0 2 , Ω=9.762ea 0 3 and Φ=22.45ea 0 4 for HCN and μ=0.1710ea0, Θ=3.2312ea 0 2 , Ω=16.578ea 0 3 and Φ=60.87ea 0 4 for HCP.

Theoretical study of the harmonic and anharmonic force fields of phosphorus pentafluoride

Abstract The harmonic force field of PF 5 is calculated at the ab initio SCF (self-consistent field) level and at the correlated MP2 level (second-order Moller-Plesset perturbation theory) using triple-zeta basis sets with multiple polarization functions (up to f functions at phosphorus). The ab initio results support the validity of a previous harmonic force field which has been refined with respect to experimental data while constraining certain off-diagonal force constants to scaled SCF values. The anharmonic force field of PF 5 is obtained from ab initio SCF calculations. The theoretical anharmonic spectroscopic constants agree reasonably well with the experimental data that are available. Many unknown spectroscopic constants are predicted.

A study of basis set and electron correlation effects in the ab initio calculation of the electric dipole hyperpolarizability of ethene (H2C=CH2)

We report finite-field self-consistent field (SCF) and many-body perturbation theory calculations of the dipole polarizability and hyperpolarizability of ethene. Large Gaussian-type basis sets have been used to ensure the determination of reliable, near Hartree–Fock values for all the independent components of the dipole hyperpolarizability tensor γαβγδ. The calculated SCF values of γ display strong basis set dependence. Electron correlation changes drastically the SCF picture of γαβγδ. The most important effect is the increase by nearly 70% of the component parallel to the double bond, γzzzz (molecular plane xz). It is found that the contribution from the fourth-order triple substitutions T4, dominates the fourth-order correction to the SCF value for almost all components of γ. The best SCF values obtained in this work are (in atomic units), γxxxx=3466, γyyyy=11 080, γzzzz=3351, γxxyy=2390, γyyzz=2936, γzzxx=1660, and γ̄=6374e4a04Eh−3. The estimated electron correlation corrections (Δ corr) are 431±79, −870±421, 2318±87, −134±101, −67±133, 240±62, and 392±236e4a04Eh−3, respectively. Albeit electron correlation affects strongly the individual tensor components of γ, the overall effect on the mean hyperpolarizability γ̄ does not exceed a few percent. The final estimate of γ̄=(68 ± 2)×102 e4 a04 Eh−3 is in essential agreement with the experimental results of Ward and Elliott.

Hyperpolarizability of H2O

A complete description of the electric dipole moment (μ), the dipole polarizability (α), the first dipole (β), and the second dipole (γ) hyperpolarizability tensors is reported for the ground state of the water molecule at its equilibrium geometry. Self-consistent-field (SCF) and complete fourth-order many-body perturbation theory (MP4) values of the independent components are calculated via a finite-field method from the perturbed energies of the molecule in the presence of a homogeneous electric field. The dependence of the calculated values on the basis set is studied at both the SCF and the MP4 levels. Electron correlation has a strong effect on the hyperpolarizability. Our best SCF values are calculated with a large (13s10p6d2f/9s6p2d)[9s7p6d2f/6s5p2d] basis set comprising 136 contracted Gaussian-type functions and are 0.7789 ea0 for the dipole moment and 8.531 e2a20E−1h, −10.86 e3a30E−2h, and 979 e4a40E−3h for the mean dipole polarizability and first and second dipole hyperpolarizabilities, respectively. The electron correlation correction to these properties is estimated at −0.055±0.005 ea0, 1.11±0.14 e2a20E−1h, −7.1±1.3 e3a30E−2h, and 749±113 e4a40E−3h. Agreement with experiment is very good for the dipole moment and mean dipole polarizability. As regards the hyperpolarizability, satisfactory agreement with the frequency-dependent values of Ward and Miller may also be deduced, but further experimental and theoretical work on the dispersion of the hyperpolarizability is needed for an effective rapprochement of theory and experiment.

A coupled cluster study of As2

Theoretical calculations at the self-consistent field (SCF) and coupled-cluster method including all single and double excitations (CCSD) have been carried out for As2. A variety of basis sets employing both segmented contractions and atomic natural orbitals have been developed and investigated. The predicted equilibrium bond length (re=2.107 Å), harmonic vibrational frequency (ωe=447 cm−1), and ionization potentials (9.75, 10.08, and 14.78 eV) are in excellent agreement with experimental values 2.103 Å, 430 cm−1, and 9.89 eV, 10.22 eV, and 15.32 eV; respectively. The best theoretical estimates for the dissociation energy (71.6 kcal/mol) and for the electron affinity (0.56 eV) were obtained at the CCSD level of theory employing large basis sets. However, given the importance of correlation effects on these two properties (the corresponding SCF values are 12.4 kcal/mol and 0.42 eV), the theoretical predictions could substantially be affected by the inclusion of connected triple excitations not considered in this work. The open-shell atomic As(4S) and diatomic calculations for As+2 and As−2 were carried out using a recently developed CCSD program that is based on restricted Hartree–Fock references and does not include any spin contamination for the correlation energies. The importance of correlating 3d electrons and the basis-set superposition error for De are analyzed. The effect of including d- and f-type polarization functions in the basis set is also discussed.

Accurate ab initio prediction of molecular geometries and spectroscopic constants, using SCF and MP2 energy derivatives

The evaluation of analytic gradients and higher derivatives has had a tremendous impact on quantum chemistry. Although self-consistent field (SCF) first derivatives and higher derivatives are now routinely evaluated as ‘black-box’ procedures, derivatives of correlated wavefunctions are not. It is argued in the following that the least sophisticated correlated method, Moller–Plesset second-order perturbation theory (MP2), can be treated so efficiently that first and second derivatives can be viewed as a ‘black-box’ procedure with the same limitations as the SCF first and second derivatives themselves.Discussed in this paper are the methods for MP2 geometry optimisation, evaluation of MP2 harmonic frequencies and prediction of i.r. intensities. Recent progress is reported which enables calculations with very large basis sets. Such calculations are reported for H2O, NH3 and CH4, where it is found that the MP2 large-basis ‘limit’ gives bond lengths within 0.003 A and bond angles within 0.5° of experimental values. Harmonic frequencies at this MP2 ‘limit’ are typically within 2% of experimental values, and i.r. intensities are much improved over SCF values. A previously suggested problem concerning the CH4 equilibrium bond length is also discussed and resolved.The evaluation of analytic third and fourth derivatives of the SCF energy is also discussed. Calculations with a DZP basis are reported for H2O, NH3 and CH4, and values are given for vibration–rotation constants α(B)r, anharmonic constants xrs, vibrationally averaged corrections to bond lengths rg–re, and also (νr–ωr), the difference between fundamental and harmonic frequencies. For nearly all these values there is good agreement with the available experiment data (typically within 10%).The conclusion of this work is that ab initio chemistry can now provide very accurate predictions of equilibrium geometries and harmonic frequencies using Moller–Plesset second-order perturbation theory (MP2). Furthermore, SCF cubic and quartic force constants can provide a good description of the anharmonic potential. Together these theoretical techniques yield an accurate full description of quadratic, cubic and quartic interactions, providing the high-resolution spectroscopist with important data on a very wide class of molecules. In addition, as the theoretical techniques are suitably formulated for vector processing, the potential applications using supercomputers instead of scalar mainframes moves from small to large molecules.