Multiconfigurational Self-consistent Field Wave Functions Research Articles

Excited-state polarizabilities of solvated molecules using cubic response theory and the polarizable continuum model

The theory and an implementation of the solvent contribution to the cubic response function for the polarizable continuum model for multiconfigurational self-consistent field wave functions is presented. The excited-state polarizability of benzene, para-nitroaniline, and nitrobenzene has been obtained from the double residue of the cubic response function calculated in the presence of an acetonitrile and dioxane solvent. The calculated excited-state polarizabilities are compared to results obtained from the linear response function of the explicitly optimized excited states.

An improved MCSCF method

An improved scheme for calculating multiconfiguration self-consistent field wavefunctions has been developed. This scheme is powerful in its algebraic formulation and quadratic in its convergence behavior. Shell and wavefunction replacement operators are introduced to make equation manipulation more transparent and less tedious. The configuration state function coefficients as well as the orbital expansion coefficients are improved in each iteration by exponential unitary transformations. The symmetry properties of the wavefunctions are fully exploited. The present scheme should be convergent for a wider range of applications than currently available schemes.

Quantum algorithm for obtaining the energy spectrum of molecular systems

Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schrödinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.

Addition of POSS−T8to the Si(100) Surface

The interaction of the polyhedral oligomeric silsesquioxane (POSS) T8, (HSiO3/2)8 with the Si(100) surface has been studied using a cluster model, represented by the single dimer Si9H12 to represent the surface. Hartree−Fock (HF), density functional theory (DFT), and multi-configuration self-consistent field (MCSCF) wavefunctions have been used to optimize the geometries of stationary points on the potential energy surface. The HF and MCSCF wavefunctions are augmented by second-order perturbation theory for improved relative energies. The POSS reacts with the Si(100) surface saturating dangling bonds on the surface, yielding two possible addition products. One of these forms a Si−Si bond between the POSS and the dimer at the same time that a hydrogen atom is transferred from the POSS to the adjacent silicon atom of the same dimer, with no structural modification of the cage. The other product involves the opening of the POSS cage with the corresponding formation of a pair of Si−Si and Si−O bonds with the ...

Nuclear magnetic resonance shielding constants in XH4 group XIV hydrides§

Self-consistent field and multiconfigurational self-consistent field wave functions are used to analyse NMR shielding constants in the XH4 hydrides, X = C, Si, Ge, Sn and Pb. All relativistic corrections to order α4, where α is the fine structure constant, are included in the evaluation of the perturbation corrections to the non-relativistic shielding constants. Each of the relativistic corrections is compared to the results obtained for group XVI and XVII hydrides and noble-gas atoms. For the heavy nuclei, the computed relativistic corrections can be used to improve the absolute shielding scale. §Dedicated to Professor Andrzej J. Sadlej on the occasion of his 65th birthday. We are pleased and happy to contribute to this volume; Happy Birthday Andrzej!

Gauge-origin-independent magnetizabilities of solvated molecules using the polarizable continuum model

We present an implementation of the polarizable continuum model in its integral equation formulation for the calculation of the magnetizabilities of solvated molecules. The gauge-origin independence of the calculated magnetizabilities and the fast basis set convergence are ensured through the use of London atomic orbitals. Our implementation can use Hartree-Fock and multiconfigurational self-consistent-field (MCSCF) wave functions as well as density-functional theory including hybrid functionals such as B3LYP. We present the results of dielectric continuum effects on water and pyridine using MCSCF wave functions, as well as dielectric medium effects on the magnetizability of the aromatic amino acids as a model for how a surrounding protein environment affects the magnetizability of these molecules. It is demonstrated that the dielectric medium effects on the magnetizability anisotropies of the aromatic amino acids may be substantial, being as large as 25% in the case of tyrosine.

Two-electron properties for the beryllium atom from explicitly correlated wavefunctions

Two-body properties of the ground state of beryllium atom in position and momentum spaces are obtained starting from a compact and accurate explicitly correlated wavefunction. The wavefunction is factorized into a correlation factor, depending on the interelectronic distance, and a multi configuration expansion. The variational Monte-Carlo method (VMC) has been used to optimize the trial wavefunction and to compute the different properties. The results are compared with others calculated from large configuration interaction expansions, multi configuration self-consistent field wavefunctions and explicitly correlated gaussian expansions, showing good performance of the results of this work. The use of explicitly correlated wavefunction along with the VMC provides reliable results for complex atomic systems as the beryllium atom.

Fine and hyperfine structure in three low-lying 3Σ+ states of molecular hydrogen

The fine structure constant (electron spin-spin coupling) and the hyperfine structure parameters (electron-nuclear spin coupling, including spin-rotation and electron-nuclear quadrupole coupling) in the low-lying triplet states b3Σ+ u, a3Σ+ g and e3Σ+ u of molecular hydrogen and deuterium are calculated using a recently developed technique with full configuration interaction and multiconfiguration self-consistent field wave functions. The second-order spin-orbit coupling contribution to the 3Σ+ states splitting is negligible, and the calculations therefore provide a good estimate of the zero-field splitting based only on the electron spin-spin coupling values. For the bound a3Σ+ g state a negligible zero-field splitting is found, in qualitative agreement with the e-a spectrum. The zero-field splitting parameter is considerable for the repulsive b3Σ+ u state (≃1 cm−1) and of intermediate size for the bound e3Σ+ u state. The isotropic hyperfine coupling constant is very large not only for the valence b3Σ+ u state (1580 MHz) but also for the Rydberg a and e triplet states (≃1400 MHz). The quadrupole coupling constants for the deuterium isotopes are negligible (0.04–0.07 MHz) for all studied triplet states. The electric dipole activity of the spin sublevels in the triplet-singlet transitions to the ground state is estimated by means of the quadratic response technique.

Ab initio study of nonhomogeneous broadening of the zero-field splitting of triplet guest molecules in diluted glasses

Nonhomogeneous broadening of phosphorescence lines and microwave signals in optical detection of magnetic resonance (ODMR) has been calculated using multiconfigurational self-consistent field wave functions and the polarized continuum model. The solvent effects on the zero-field splitting (ZFS) parameters in the low-lying triplet states of aza-aromatic molecules are found to be linearly dependent on the solvent-induced shifts in the phosphorescence frequency in agreement with experimental data. The main contribution to the ZFS originates in the dipolar interaction of the two electron spins, the spin–spin coupling. The spin–orbit coupling (SOC) contribution to the ZFS parameter is much larger for the 3nπ* state of pyrazine compared to the 3ππ* states of quinoline. The second-order SOC contribution to the splitting of the 3nπ* state in the pyrazine molecule does not show any appreciable dependence on the dielectric constant of the solvent. This raises doubts about earlier theories for explaining the inhomogeneous broadening in triplet-state spectra based exclusively on the SOC-induced mixing of the singlet and triplet states. We complete the interpretation of the ODMR spectrum of pyrazine by calculating the hyperfine coupling (HFC) tensors in the lowest triplet state using the UB3LYP hybrid functional. An appreciable solvent-induced rotation of the anisotropic HFC tensor axes has been obtained for the 3nπ* state of pyrazine, in particular for 13C and 14N nuclei. This produces additional nonhomogeneous broadening not only in electron–nuclear double resonance spectra, but also in electron paramagnetic resonance signals because the anisotropic HFC perturbation results in an intensity redistribution among the magnetic transitions between the spin sublevels. A small in-plane rotation of the ZFS tensor axes upon solvation has been predicted for quinoline. Rotation of the magnetic axes induced by the interaction with isotropic solvents can provide a new mechanism for spin-lattice relaxation in the triplet state.

Quantum Monte Carlo study of singlet–triplet transition in ethylene

A theoretical study is reported of the transition between the ground state (1Ag) and the lowest triplet state (1 3B1u) of ethylene based on the diffusion Monte Carlo (DMC) variant of the quantum Monte Carlo method. Using DMC trial functions constructed from Hartree–Fock calculations, complete active-space self-consistent field and multiconfiguration self-consistent field wave functions, we have computed the atomization energy and heat of formation of both states and the adiabatic and vertical energy differences between these states using both all-electron and effective core potential DMC methods. The ground-state atomization energy and heat of formation are found to agree with experiment to within the error bounds of the computation and experiment. Predictions by the DMC method of the triplet-state atomization energy and heat of formation are presented. The adiabatic singlet–triplet energy difference is found to differ by 5 kcal/mol from the value obtained in a recent photodissociation experiment.

A linear response approach to second-order electronic transition intensities for multiconfigurational self-consistent field wave functions

A new theoretical approach to two-photon transition intensities at the multiconfigurational self-consistent field (MCSCF) level of theory, is described in detail. The fundamental property of an MCSCF wave function, that it is possible to define the response equations for an excited state, is a prerequisite. The method requires solely first-order multiconfigurational response calculations, because the equations involve the response of both the initial and final state. However, the method is approximate as the coupling between the +ω and −ω parts of the linear response is disregarded. The complete active space state interaction (CASSI) method is applied in the evaluation of the involved matrix elements. To illustrate the performance and the requirements of this method, it was used to determine TP transitions in trans-1,3-butadiene and trans-stilbene.

Ab initio calculations of zero-field splitting parameters

We present calculations of electron spin–spin (SS) coupling strengths evaluated as expectation values over multi-configuration and restricted high-spin self-consistent field wave functions. Together with the spin–orbit (SO) configuration interaction methodology, this enables us to analyze the full spin Hamiltonian including the zero-field splitting (ZFS) parameters of the triplet state. The calculated ZFS parameters include both the SS coupling to first order and SO coupling to second order of perturbation theory. The relative importance of these two contributions is strongly system dependent. In the lowest triplet state of the benzene molecule, the main ZFS parameter – the D parameter – is determined entirely by the SS coupling, with D calculated to be 0.1583 cm −1 . In contrast, the calculated D parameter in the X 3Σ g − ground state of the oxygen molecule ( 3.77 cm −1 ) includes a large one-center SS contribution (D SS =1.455 cm −1) but an even larger SO coupling contribution ( D SO =2.315 cm −1 ). ZFS parameters for molecular oxygen excited states, A 3Σ u + and B 3Σ u − , which belong to Hund's case (b), are also calculated. A large negative D value (−10.2 cm −1) for the A 3Σ u + state is to 90% determined by the D SO contribution, while the Schumann–Runge state spin splitting is mainly determined by SS coupling. The calculated values for benzene and oxygen are in good agreement with data from EPR and rotational fine-structure spectra. The applicability of response theory is with this contribution expanded to include the calculation of the SS coupling of the spin Hamiltonian, complementing the previous implementations of hyperfine A- and g-tensors.

Optimization of MCSCF excited states using directions of negative curvature

A line search method that uses directions of negative curvature for the optimization of ground and excited state multiconfigurational self-consistent field (MCSCF) wavefunctions is suggested. The method is applicable to general MCSCF wavefunctions and not restricted to specific classes of model spaces, such as CASSCF functions. It is shown that the approach can be implemented with Newton or quasi-Newton methods for determination of descent directions. Thus, the method is particularly promising for cases in which it is inconvenient or costly to calculate exact curvature matrices. We demonstrate the viability of the approach by numerical example on the difficult BeO problem.

Electronic g-tensors obtained with the mean-field spin–orbit Hamiltonian

The accuracy of the atomic mean-field approximation (AMFI) to the spin–orbit interaction Hamiltonian is evaluated for the paramagnetic contribution to electronic g-tensors. A variety of different substances are tested for this purpose for restricted open-shell Hartree–Fock and multi-configuration self-consistent field wave functions. In most cases the introduced error is significantly lower than errors of wave function parameterization. Considering the substantial computational savings, it is argued that the mean-field approximation is warranted given the resolution available in current electron spin resonance experiments.

Intrinsic bond strengths of multiple gallium–gallium bonds: A compliance matrix study using multiconfiguration self-consistent field wave functions and hybrid density functionals

Inverted matrices of force constants (compliance matrices) are used to reassign the bond order of the Ga–Ga bond in a recently synthesized compound Na2{GaC6H3–2,6-Trip2}2 (Trip=C6H2–2,4,6–Pr3). The results of multiconfiguration wave functions using the complete active space self-consistent field method are compared to hybrid density functionals and Hartree–Fock wave functions.

The Cotton–Mouton effect of gaseous CO2, N2O, OCS, and CS2. A cubic response multiconfigurational self-consistent field study

The hypermagnetizability and the hypermagnetizability anisotropy of CO2, N2O, OCS, and CS2 are computed at a wavelength of 632.8 nm using cubic response theory with multiconfigurational self-consistent field wave functions. The anisotropies of the electric dipole polarizability and of the magnetizability are also obtained. This allows us to study the temperature dependence of the Cotton–Mouton constant for all four molecules and thus to compare to the results of the experimental study by Kling and Hüttner [Chem. Phys. Lett. 90, 207 (1984)]. We also assess the importance of pure and zero-point vibrational effects on the relevant molecular properties. In particular, we show that for CO2, OCS, and CS2, the pure vibrational effects to the hypermagnetizability anisotropy can be even more important than the electronic contribution.

Electron propagator method with a multiconfigurational second-order perturbation theory wave function as the initial state in the fermion operator block

We have developed an electron propagator method using a multiconfigurational second-order perturbation theory (CASPT2) wave function as the initial state [electron propagator CASPT2 (EPCASPT2)] in the fermion operator block (block 1). In the other blocks a multiconfigurational self-consistent field wave function is the initial state. We apply our new method to directly determine the low-lying vertical ionization potentials of Be, CH2, NH2, and H2O. We compare our results with the results of the calculations using multiconfigurational spin tensor electron propagator (MCSTEP), full configuration interaction (FCI), and multireference configuration interaction (MRCI) methods with the same geometries and basis sets. The calculations are performed using complete active space (CAS) choices that are usually excellent for MCSTEP ionization potential (IP) calculations and also for CAS choices that are inadequate for MCSTEP IP calculations. We show that EPCASPT2 generally improves MCSTEP IPs compared to ΔFCI when the MCSTEP IPs are in very good to excellent agreement with ΔFCI IPs and that EPCASPT2 can effectively mimic ΔFCI even when the CAS choice for the initial state is inadequate for MCSTEP.

MCSCF calculations of NMR spin–spin coupling constant of the HF molecule

The dependence of spin–spin NMR coupling constants on the basis set and electron correlation has been investigated for the hydrogen fluoride using Hartree–Fock (HF-SCF) and multiconfigurational self-consistent field (MCSCF) wave functions. The effect of the size, contraction, and tight s-type, augmented and polarization functions in the basis sets is analyzed. MCSCF wave functions with different number of active orbitals and excited electrons were used within the frozen-core approximation and with all-electron calculations. The correlation effect associated with the core electrons is not negligible. An approximation to determine spin–spin coupling constants at high level of electron correlation and reduced computational cost is applied satisfactorily. The best calculated and estimated 1JFH couplings are 544.20 and 536.63 Hz, respectively, with all electron correlation. Both values agree with the experimental one within the error bars (525±20 Hz).

Relativistic MCSCF by means of quasidegenerate direct perturbation theory. I. Theory

Relativistic MC-SCF (multiconfiguration self-consistent field) in terms of quasidegenerate direct perturbation theory (DPT) of relativistic effects is formulated based on a recently presented theory of effective Hamiltonians for electrons in a model space. The appropriately defined diagonal and nondiagonal parts of operators play a key role in this context. Their definition is based on stationary conditions for the MC-SCF wave function. The formalism starts from nonrelativistic MC-SCF theory. The leading relativistic correction appears as an expectation value in terms of the nonrelativistic MC-SCF function, while the higher-order relativistic corrections require a coupled-MC-SCF type approach.

Rovibrationally averaged magnetizability, rotational g factor, and indirect spin–spin coupling of the hydrogen fluoride molecule

The magnetizability tensor, the rotational g factor, and the indirect nuclear spin–spin coupling constant of the hydrogen fluoride molecule have been calculated using large multiconfigurational self-consistent field wave functions and large basis sets. For a critical comparison with experiment, rovibrational corrections have also been calculated. For the magnetizability tensor and the spin–spin coupling constant, we present results with higher precision than available experimental data; for the rotational g factor, our results are in good agreement with experiment.