MCSCF Reference Research Articles

Nuclear Spin–Spin Couplings: Efficient Evaluation of Exact Exchange and Extension to Local Hybrid Functionals

We present an efficient implementation for the computation of nuclear spin-spin coupling tensors within density functional theory into the TURBOMOLE software suite. Emphasis is put on methods to efficiently evaluate the Hartree-Fock exchange needed for hybrid functionals: resolution of the identity and seminumerical evaluation on a grid. Our algorithm allows for the selection of specific nuclei for the reduction of calculation times. Further, the accuracy of locally dense basis sets in the density functional theory framework is investigated. These features allow for the routine computation of coupling constants in systems comprising about 100 carbon atoms within less than one day on a single CPU and within a few hours when using the OpenMP variant. Based on seminumerical integration, the first implementation of local hybrid functionals for spin-spin couplings is reported. This has allowed a preliminary evaluation of position-dependent exact-exchange admixture in three local hybrid functionals for a set of 80 isotropic spin-spin couplings in 23 small main-group molecules against CC3 and MCSCF reference data. Two of the local hybrids (LH14t-calPBE and LH07t-SVWN) are the top performers in the overall statistical evaluation compared to several standard functionals (TPSS, TPSSh, B3LYP, PBE0, and BHLYP), in particular, as they do not exhibit notable outliers for specific coupling types.

The evaluation of MCRPA (MCTDHF) electronic excitation energies, oscillator strengths, and polarizabilities: Application to O2

We derive the multiconfigurational random phase approximation (MCRPA) by requiring the equivalence between the length and velocity forms of the oscillator strength. The MCRPA is also justified in terms of the equations of motion (EOM) formalism. The final equations and techniques developed are the same as the multiconfigurational time dependent Hartree-Fock (MCTDHF). We explicitly demonstrate the validity of the energy-weighted sum rule for the MCRPA-MCTDHF. A modification of our previously derived MCTDHF technique is introduced. This involves the incorporation of additional CI states from different symmetry blocks than the symmetry block of the MCSCF reference state into the MCRPA equations. For a 50 STO calculation on O2 molecule, our results for the low-lying vertical excitation energies are in excellent agreement with experiment and with large-scale CI. Oscillator strengths, frequency dependent polarizabilities, and the energy weighted sum rule are also reported.

Combining two-body density functionals with multiconfigurational wavefunctions: diatomic molecules

The MCSCF method provides a correct zero-order wavefunction for all regions of molecular potential energy surfaces. To obtain quantitative accuracy a proper treatment of the dynamic correlation problem must be implemented. Traditionally this has been achieved through multireference variants of perturbation theory, configuration interaction and coupled cluster theory. The computational cost of such techniques makes them prohibitive for all but the smallest molecular problems. Reported here is an investigation into the efficacy of two-body density functionals in providing the dynamic correlation energy for MCSCF reference states. Tests were made on the two-body density functionals of Colle and Salvetti (CS), Moscardó and San-Fabán (MSF), and Moscardó and Pérez-Jiménez (MPJS) in predicting the equilibrium bond lengths, harmonic frequencies and dissociation eneries of fifteen diatomic molecules (3B2, 3BN, 2BS, 1C2, 2CN, 1CO, 1F2, 1FC1, 1N2, 3NCl, 3O2, 1PN, 3Si2, 1SiO, 3SO) using full valence-shell CASSCF reference wavefunctions. Also studied were modifications of these functionals recently suggested by Miehlich, Stoll and Savin (MSS) and Gráfenstein and Cremer (GC). The results obtained show accuracy comparable with and typically superior to the popular Kohn-Sham BLW and B3LW methods. However, the latter methods are not applicable in all regions of a potential energy surface, and even predict incorrect ground states for some systems. The use of two-body density functionals with MCSCF reference states does not share this shortcoming.

MCSCF reference quasidegenerate perturbation theory with Epstein—Nesbet partitioning

A quasidegenerate perturbation theory (QDPT) with Epstein—Nesbet partitioning based on MCSCF reference functions is presented. The formulas for CSF-based matrix operations are derived. They are applied to the systems BeH 2 and CO, and compared with other methods. The potential energy curves agree well with those of the full-CI or MR-CI methods.

Calculation of barriers to proton transfer using a variety of electron correlation methods

The usefulness of various combinations of MCSCF and CI methods in computing correlated proton transfer potentials is investigated for the systems, HF, H7N, H3O, and H5O. MCSCF calculations can accurately determine proton transfer barriers, provided the correlation is limited to the proton transfer process. The proper correlated space can be obtained more easily if the canonical occupied MOs are first subjected to a localization. Various means are tested of including additional electron correlation into the MCSCF methods. CIS and CISD calculations are performed following MCSCF expansion of the wave function using various different MCSCF reference wave functions. The MCSCF + CISD results are excellent, being fairly independent of choice of virtual Mos, although it is important that the occupied orbitals be balanced between the donor and acceptor. Localizing the occupied MOS prior to the MCSCF part of the calculation again results in a further improvement. These results are compared to CI computations using the canonical orbitals (and which are not preceded by MCSCF preparation of the wave function) and to Møller–Plesset results. © 1992 John Wiley & Sons, Inc.

First-order correlation orbitals for the MCSCF zeroth-order wave function

The originally proposed first-order correlation orbital method (L. Adamowicz and R.J. Bartlett, J. Chem. Phys. 86 (1987) 6314) has been extended in order to include the MCSCF zeroth-order wave function. The present method is based on the perturbation theory with the zeroth-order Hamiltonian suggested by Andersson (J. Phys. Chem. 94 (1990) 5483). In the present work we restrict our consideration to the simplest case, i.e. the two-electron/two-orbital MCSCF reference wave function. The procedure is tested on the model system — lithium dimer — at a variety of interatomic distances. The results show that the calculated second-order correlation energy as well as the value of the second-order Hylleraas functional in the reduced space of the first-order correlation orbitals behave correctly upon dissociation of the system.

Generalized Møller—Plesset perturbation theory applied to general MCSCF reference wave functions

We report the application of a generalized Møller—Plesset perturbation theory to multiconfigurational reference wave functions. The theory makes use of a more compact first-order space than earlier investigations and results in a computationally simplified approach. The low-order perturbation theory applied to multiconfigurational reference functions obtains a significant fraction of dynamic correlation energy for a relatively low computational cost. The importance of using a reference wave function that dissociates properly is discussed. Results are reported for the symmetric dissociation of H 2O and the energy difference between the 1A 1 and 3B 1 states of CH 2.

A theoretical study of the complexes of N2O with H+, Li+, and HF using various correlation methods

Binding energies for complexes of N2O with the acids H+, Li+, and HF have been computed using the following correlation methods: many-body (Moller–Plesset) perturbation theory at second (MP2), third (MP3), and fourth (MP4) order; the “quadratic CI” method with single and double excitations (QCISD) and with noniterative inclusion of triple excitations (QCISD(T)); the linearized coupled-cluster method (LCCM); the averaged coupled-pair functional (ACPF); configuration interaction with all single and double excitations (CISD); and CISD with the Davidson and Pople corrections. The convergence of the Moller–Plesset expansion is erratic, predicting that the terminal nitrogen is the preferred binding site for the complexes at the MP2 and MP4 levels, in disagreement with Hartree–Fock and MP3 and all other models (including the infinite-order QCI). The effect of triple excitations at MP4 and QCI is to destabilize complexes bound at O and stabilize those bound at N, but this effect is greatly overestimated at MP4 relative to QCI. Except for the LCCM result for N-protonated N2O, ACPF and LCCM bindin energies are similar to the QCISD values. The size-consistency error in the ACPF binding energies of the complexes of N2O with HF is about 0.5 kcal/mol. The CISD size-consistency error for these complexes is 23 kcal/mol, leading to negative binding energies when computed relative to isolated N2O and HF. The Davidson correction reduces the size-consistency error but still leaves the binding energies negative. The Pople correction produces positive but too small binding energies. However, all methods give consistenly good, essentially indistinguishable binding energies when computed relative to a similar calculation on a supermolecule of infinitely separated N2O and HF units. Multireference ACPF and CISD binding energies have also been obtained using MCSCF reference functions which correlate the highest occupied and lowest virtual pairs of π orbitals. The multireference binding energies are not greatly different from the single-reference values, and do not reverse the negative binding energies of the complexes with HF at the CISD level when computed relative to isolated molecules. When calculated relative to the infinitely separated supermolecule, the multireference binding energies of these complexes are virtually identical to the corresponding single-reference values.

Application of the multiconfiguration time-dependent Hartree–Fock method to CH+: The auxiliary active space

The multiconfigurational time-dependent Hartree–Fock (MCTDHF) method is used to study the excitation energies and transition moments of the CH+ ion. A consistent approach to the selection of configurations based on the complete active space (CAS) model is successfully applied. In this approach a large set of orbitals is included in the active space without a substantial increase in the number of configurations, since only for a smaller subset of orbitals is the configuration expansion complete and the remaining orbitals active are considered an auxiliary set. For CH+ the active space includes the valence and additional orbitals, with the valence orbitals constituting the complete subset. Extra configurations involving single and double excitations from the valence to the additional (auxiliary) orbitals are included both for the MCSCF reference state and in the MCTDHF. It is demonstrated that this active space accurately mimics MCSCF and MCTDHF calculations with an enlarged complete active space which includes both the original CAS and the auxiliary orbitals. The calculations for the 1 Π and 1 Σ+ excited states at the equilibrium geometry show that accurate excitation energies and transition moments are obtained in this way. For most of these states for which there are other good ab initio results available, we obtain comparable excitation energies and transition moments. In addition to 1 Π and 1 Σ+, low lying states of other symmetries are obtained and discussed.

On the evaluation of non-Born–Oppenheimer interactions for Born–Oppenheimer wave functions. V. A body fixed frame approach. Applications to isotope effects on equilibrium geometries and the adiabatic correction for the X 1Σ+ state of LiH

The evaluation of the total second derivative nonadiabatic coupling matrix element H(J,I,R)=〈ψJ(r;R)‖∑i(−1/2 Mu)(∂2/∂R2i )ψI(r;R)〉r is considered. Here ψJ(r;R) is the adiabatic Born–Oppenheimer electronic wave function which in this work will be approximated by a large-scale CI wave function developed from an MCSCF reference space. For diatomic and triatomic systems the computational effort associated with the evaluation of H(J,I,R) can be reduced considerably by the use of a body fixed frame approach. In this approach costly evaluation of the derivative wave function with respect to noninternal degrees of freedom in the space fixed frame is replaced by the evaluation of matrix elements of many electron operators including the mass polarization operator (total electronic linear momentum squared) and the L2 operator (total electronic orbital angular momentum squared). The equivalence of the body fixed frame and space fixed frame results leads to valuable diagnostic equations which provide stringent tests of the derivative methodology used to evaluate the remaining second derivatives with respect to internal coordinates. The methods presented here are applied to the benchmark systems BeH+ and LiH. The Born–Oppenheimer diagonal correction or adiabatic correction (AC) is evaluated for the X 1∑+ state of these systems and used to consider the effect of isotopic substitution on equilibrium geometries. For the X 1∑+ state of LiH a troubling discrepancy exists between the AC determined by advanced theoretical and experimental techniques. For R≲Re the AC determined directly with specialized CI wave functions and the experimental value inferred from a detailed spectroscopic analysis of the A→X emission agree. However, for R>Re theory and experiment disagree qualitatively. For R≲Re our results are consistent with the previous work. For R>Re our results are in accord with the experimentally derived AC.

The electronic and molecular structure of silyl nitrene

FORS MC SCF calculations performed with the 6-31G d basis set have been used to predict the geometries of the lowest singlet ( 1E) and triplet ( 3A 2) states of silyl nitrene. Both states are found to have C 3v symmetry at this computational level. Single- second-order configuration interaction calculations that is, all single and double excitations from the FORS MC SCF reference configuration) at the MC geometries predict the triplet ground state to be 42.3 kcal/mol below 1E and only 8.2 kcal/mol above the singlet ground state of silaimine.

The ionization potentials of NH2: The multiconfigurational spin-tensor electron propagator method (MCSTEP) applied to a polyatomic open-shell radical

The multiconfigurational spin-tensor electron propagator method (MCSTEP) gives accurate ionization potentials (IPs) and electron affinities (EAs) for both closed-shell and open-shell molecules, including for highly correlated systems. Both principal and lower-lying shakeup IPs can be accurately obtained and straightforwardly characterized using MCSTEP. To further test this new technique, we have applied the MCSTEP approach to the open-shell, polyatomic radical NH2. We report and characterize vertical IPs 0–20 eV, including several ionizations that have not, as yet, been observed experimentally. IPs to both singlet and triplet states of NH+2 are accurately calculated using the same MCSCF reference state. We predict the presence of previously undetected, observable vertical PES IPs to states of 3A2, 1A1, and 1A2 symmetries at 16.86, 18.00, and 18.26 eV, respectively. Also, we calculate adiabatic IPs by a modified application of the procedure. By examining the two lowest adiabatic IPs with both MCSTEP and Δmultireference singles+doubles CI at geometries established by Pople’s research group and with the basis set used by that group augmented by hydrogen d polarization functions we conclude that their MP4 procedures overestimate the IPs with their best basis set by ∼0.2–0.3 eV.

The radiative lifetime of A 1Πu C2

This note reports ab initio potential curves of A 1Πu and X 1Σ+g states of C2, and the A–X radiative lifetimes. The calculations are based upon configuration interaction wave functions formed from single and double electron excitations from state-averaged MC-SCF reference functions expanded in a Gaussian basis. The lifetimes for (J′=1,v′) A 1Πu are calculated to be about 25% longer than those found in previous ab initio works, but about 25% shorter than the most recent LIF measurements of Bauer, Becker, Bielefeld, and Meuser. Also reported are the quadrupole moments of the A 1Πu and X 1Σ+g states as a function of internuclear distance and the quadrupole lifetimes of several (v′=0,J′) levels in the X state.

Bound electronic states of HCl−

This paper reports ab initio Born–Oppenheimer calculations on the X 1Σ state of HCl and on the 1 2Σ, 2 2Σ, and 2Π states of HCl− for internuclear distances at which these states are electronically bound. The calculations employed square-integrable configuration interaction wave functions which included all single and double replacements from multiconfiguration reference functions. The MC-SCF reference configurations and the Gaussian atomic basis encompass the dominant effects which govern the electronic states studied. In the process of these calculations, we have calculated the static polarizability of Cl− and reproduced the known polarizabilities of H, Cl, and H−. The calculations on the X 1Σ state of HCl yield a De within 0.10 eV of the measured value, and ωe within 3 cm−1 of the measured value. For the molecular ion, we find the potential of the 1 2Σ state of HCl− to be attractive and to undergo substantial charge transfer, and the potentials of the 2Π and 2 2Σ states to be repulsive even at large internuclear separations.

Excitation energies in Be: A comparison of multiconfigurational linear response and full configuration interaction calculations

Using a 〈9s9p5d〉 contracted GTO basis we have calculated low-lying excitation energies of singlet and triplet symmetry for the Be atom using Δfull CI, ΔCI(1s) with double occupancy in the 1s orbital, multiconfiguration linear response (MCLR), and ΔMCSCF approaches. The Δfull CI results agree very closely with the experimental excitation energies except for higher excitations where obvious basis set defects occur. The MCLR calculations shows that with an adequately chosen MCSCF reference state the MCLR calculation is capable of mimicking the Δfull CI results. The MCLR results are closer to the Δfull CI results than the ΔCI(1s). The ΔMCSCF excitation energies show that this approach can only be used with extreme care to determine excitation energies.

Simple molecular wavefunctions with correlation corrections II

The potential curves for the ground state of Li2 (1Σ g + ) and FH (1Σ g + ) are computed. The correlation energy is calculated using a functional of the one- and two-electron density matrices derived from an MC SCF reference wavefunction and is added to the reference energy to obtain a correlated potential curve.

Linear and nonlinear response functions for an exact state and for an MCSCF state

We have examined the response of an exact and an MCSCF reference state to a general time-dependent field. The time development of both the exact and the MCSCF reference state have been parametrized in terms of explicit exponential time-dependent transformations. The time development has been determined by requiring the Ehrenfest theorem to be satisfied through each order in the interaction between the molecular system and the field. The response of the exact and the MCSCF reference state has been used to evaluate linear, quadratic, and cubic response functions. It has been shown how a large variety of molecular properties may be expressed in terms of these response functions. It has also been demonstrated that molecular properties containing the electric dipole operator may be expressed in equivalent forms involving the momentum operator both for the exact and the MCSCF state. The MCSCF response functions have been transformed to computationally attractive expressions which do not contain summation indices over intermediate states and which allow direct techniques to be straightforwardly applied.

Generalizations of the multiconfigurational time-dependent Hartree–Fock approach

We extend and generalize the time-dependent Hartree–Fock (TDHF) and multiconfigurational time-dependent Hartree–Fock (MCTDHF) approaches so that electronic transition energies from non-singlet reference states to states of pure spin symmetry other than the reference state can be described. Initial calculations are presented for the Be atom using the (2s3s)3S state and the (2s2)1S state as reference states. The accuracy of the calculated excitation energies is examined as a function of extending the active space of the MCSCF reference state. Close agreement between the excitation spectra as obtained from either of the reference states is obtained when inner–outer correlation effects are considered in the MCSCF reference state.

MCSCF–CI calculations of the ground state potential curves of LiH, Li2, and F2

The potential curves for LiH(X 1Σ+), Li2(X 1Σ+g), and F2(X 1Σ+g) have been calculated using a wave function comprising all single and double replacements from an MCSCF reference state. It is shown that in order to obtain satisfactory results, the reference function should describe properly not only the dissociation of the molecule, but also the united atom limit. This implies that for Li2, 1πu, and for F2,2πu orbitals should be included in the reference state. Calculated dissociation energies are 2.48 (2.52) eV for LiH, and 1.00 (1.05) eV for Li2 (experimental values within parentheses). For F2 only 1.31 (1.66) eV is obtained with 2πu excluded from the reference orbital space. A calculation including this orbital yields 1.40 eV even at the MCSCF level.

On the 3d24s2, 3d34s1, and 3d4 states of Ti

The 3F, 5F, and 5D(3d4) states of Ti are investigated using CI and MCSCF calculations.(AIP)