Multiconfigurational Self-consistent Field Calculations Research Articles

Electronic structure and spin coupling of the manganese dimer: The state of the art of ab initio approach

A thorough ab initio study of the Mn(2) dimer in its lowest electronic states that correlate to the ground Mn((6)S)+Mn((6)S) dissociation limit is reported. Performance of multireference methods is examined in calculations of the fully spin-polarized S=5((11) summation operator(+) (u)) state against the recent accurate single-reference coupled cluster CCSD(T) results [A. A. Buchachenko, Chem. Phys. Lett. 459, 73 (2008)]. The detailed comparison reveals a serious disagreement between the multireference configuration interaction (MRCI) and related nonperturbative results on the one hand and the complete active space perturbation theory (CASPT) calculations on the other. A striking difference found in the CASPT results of the second and third orders indicates poor perturbation expansion convergence. It is shown that a similar problem has affected most of the previous calculations performed using CASPT2 and similar perturbative approximations. The composition of the active space in the reference multiconfigurational self-consistent field calculations, the core correlation contribution, and basis set saturation effects are also analyzed. The lower spin states, S=0-4, are investigated using the MRCI method. The results indicate a similar dispersion binding for all the spin states within the manifold related to the closed 4s shells, which appears to screen and suppress the spin coupling between the half-filled 3d atomic shells. On this premise, the full set of model potentials is built by combining the accurate reference CCSD(T) interaction potential for S=5 and the MRCI spin-exchange energies for the S<5 states. This approach leads to the value of 550 cm(-1) as a lower bound for the (1) summation (+) (g) ground-state dissociation energy. The spin-exchange energies themselves are found to comply with the simple Heisenberg model. The effective spin-coupling parameter J is estimated as -3.9 cm(-1), a value roughly 2.5 times smaller in magnitude than those measured in the inert gas cryogenic matrices. Compressing of the Mn(2) dimer in the matrix cage is suggested as the prime cause of this disagreement.

Developments in the calculation of electronic wavefunctions for molecules: MCSCF, CI, and numerical SCF for molecules

The multiconfiguration self-consistent-field (MCSCF) theory will be presented, discussing in particular (a) the advantages, necessity, and limitations of the method; (b) the merits of the various different procedures for the solution of the MCSCF orbital equations, and (c) some considerations useful for the configuration selection in an MCSCF calculation. A scheme for large scale configuration interaction (CI) calculations based on a general multiconfigurational reference function will be emphasized. The progress made and experiences gained in the development of a numerical SCF procedure for general molecules will be described, giving an outlook towards promise and future extensions of this procedure.

The electronic structure of singlet cyclopentadienylidene

We have carried out ab initio multiconfigurational self-consistent field calculations on the lowest singlet state of the carbene C5H4. We find that there are, along a restricted geometry deformation coordinate, two local energy minima corresponding to localized and delocalized π bonds having 4π and 6π electrons, respectively. The role of 4n + 2 resonance in determining the nature of the electronic structure and reactivity of this carbene is analyzed and compared with that of other carbenes.

Binding and Diffusion of Al Adatoms and Dimers on the Si(100)-2 × 1 Reconstructed Surface: A Hybrid QM/MM Embedded Cluster Study

When group III metals are deposited onto the Si(100)-2 × 1 reconstructed surface they are observed to self-assemble into chains of atoms that are one atom high by one atom wide. To better understand this one-dimensional island growth, ab initio electronic structure calculations on the structures of Al atoms on silicon clusters have been performed. Natural orbital occupation numbers show that these systems display significant diradical character, suggesting that a multireference method is needed. A multiconfiguration self-consistent field (MCSCF) calculation with a 6-31G(d) basis set and effective core potentials was used to optimize geometries. The surface integrated molecular orbital molecular mechanics embedded cluster method was used to take the surface chemistry into account, as well as the structure of an extended surface region. Potential energy surfaces for binding of Al adatoms and Al−Al dimers on the surface were determined, and the former was used to obtain a preliminary assessment of the surface diffusion of adatoms. Hessians were calculated to characterize stationary points, and improved treatment of dynamic electron correlation was accomplished using multireference second order perturbation theory (MRMP2) single-point energy calculations. Results from the MRMP2//MCSCF embedded cluster calculations are compared with those from QM-only cluster calculations, embedded cluster unrestricted density functional theory calculations, and previous Car−Parrinello DFT studies.

The Binding of Ag+ and Au+ to Ethene

For the reaction M(+)(C(2)H(4))(n-1) + C(2)H(4) --> M(+)(C(2)H(4))(n), where M = Ag, Au, the binding energies are predicted at the second order perturbation (MP2) and coupled cluster (CCSD(T)) levels of theory. As the basis set is systematically improved, the predicted M = Ag binding energies steadily improve, as compared to the experimental values. In fact, the complete basis set limit (CBS) predicted CCSD(T) binding energy for Ag(C(2)H(4))(+) is within experimental error. For MP2, as the basis set is improved, the agreement with experiment worsens. Gold ions are predicted to bind more strongly than silver ions to ethene ligands. Mulliken population analyses of the silver and gold systems exhibit delocalization of the positive charges of the metal ions onto the ethene ligands. Reduced variational space analysis indicates that electrostatic interactions are the principal contributor to the bonding in these systems. Multiconfigurational self-consistent field calculations do not support the Dewar-Chatt-Duncanson model of transition metal-alkene bonding in Au(C(2)H(4))(+).

Excitation energy and pair correlation function of trions near an LiF surface

We investigate the excitation spectrum of a singly charged LiF surface using an embedded cluster approach. The spectrum is determined by multiconfiguration self-consistent field calculations. We find stable trionic states, i.e., quasimolecular states of a three-quasiparticle complex consisting of two holes and one electron interacting through the Coulomb potential. The excitation energy of the lowest trionic state is approximately 12.1 eV. The mean hole-hole spacing is about 7.6 a.u. (or $4\text{ }\text{\AA{}}$). Our results confirm the interpretation of experiments [P. Roncin et al., Phys. Rev. Lett. 89, 043201 (2002)] where trion excitation was suggested as effective energy-loss channel during the neutralization of ${\text{Ne}}^{+}$ ions at an LiF surface.

Approximating correlation effects in multiconfigurational self-consistent field calculations of spin-spin coupling constants

The multiconfigurational self-consistent field (MCSCF) method in their approximations restricted and complete active spaces (RAS and CAS) provides a theoretically accurate description of the coupling constants of a wide range of molecules. To obtain accurate results, however, very large basis sets and large configuration spaces must be used. Nuclear magnetic resonance coupling constants for the equilibrium geometry have been calculated for a series of small molecules using approximated correlation contributions. The four contributions to the coupling constants (Fermi contact, spin dipolar, orbital paramagnetic, and orbital diamagnetic) have been calculated at the CAS and RAS MCSCF and second-order polarization propagator approximation levels using a large basis set. An additive model that considers the effect on the coupling constants from excitation of more than two electrons and from core-electron correlation is used to estimate the coupling constants. Compared with the experimental couplings, the best calculated values, which correspond to the MCSCF results, present a mean absolute error of 3.6 Hz and a maximum absolute deviation of 13.4 Hz. A detailed analysis of the different contributions and of the effects of the additive contributions on the coupling constants is carried out.

A CASSCF and CASPT2 study of the photochemistry of 1,1- and 1,2-difluoroethylenes

Multiconfigurational complete active space self-consistent field (CASSCF) calculations corrected with second order perturbation theory (CASPT2) have been employed to characterize the valence and 3s and 3p Rydberg states of 1,1- and 1,2- cis and trans difluoroethylenes. The calculated energies show a good agreement with the available experimental values. Potential energy curves along the torsion around the double bond and the pyramidalization at both carbons have been computed to gain insight into the possible crossing points between the valence and Rydberg states. Whereas few changes are found in the Rydberg states along these two coordinates, a number of degenerated points are found between the valence states; they serve as starting points for the optimization of conical intersections. Conical intersections for torsion, pyramidalization and hydrogen migration have been located. Based on these stationary points, the photodissociation mechanisms of difluoroethylenes is discussed.

Equivalent orbitals for multiconfigurational spin‐tensor electron propagator method (MCSTEP): The vertical ionization potentials of B, NO, CF, and OF

AbstractThe multiconfigurational spin tensor electron propagator method (MCSTEP) was developed as an implementation of electron propagator/single particle Green's function methods for ionization potentials (IPs) and electron affinities (EAs). MCSTEP was specifically designed for open shell and highly correlated (nondynamically correlated) initial states. For computational efficiency the initial state used in MCSTEP is typically a small complete active space (CAS) multiconfigurational self‐consistent field (MCSCF) state. If in a molecule there are some degenerate orbitals which are not fully or half occupied, usual MCSCF calculations will make these orbitals inequivalent, i.e., the occupied ones will be different from the nonoccupied ones, so that the degeneracy is broken. In this article, we use a state averaged MCSCF method to get equivalent orbitals for the initial state and import the integrals into the subsequent MCSTEP calculations. This gives, in general, more reliable MCSTEP vertical IPs. © 2008 Wiley Periodicals, Inc., 2008

Multireference self-consistent-field energies without the many-electron wave function through a variational low-rank two-electron reduced-density-matrix method

The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17x10(9) determinants in the active space, could not be performed.

Multireference many-electron correlation energies from two-electron reduced density matrices computed by solving the anti-Hermitian contracted Schrödinger equation

Two-electron reduced density matrices (2-RDMs) have recently been directly calculated by solving the anti-Hermitian contracted Schr\"odinger equation (ACSE) to obtain $95$--$100\phantom{\rule{0.3em}{0ex}}%$ of the ground-state correlation energy of atoms and molecules with the accuracy increasing with the size of the one-electron basis set [Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).] In this paper, the ACSE method is extended to treat strong multireference correlation effects that are often important at nonequilibrium molecular geometries. While previous ACSE calculations have employed an initial 2-RDM from the Hartree-Fock method, we initialize the solution of the ACSE with a 2-RDM guess from a multiconfiguration self-consistent field calculation. Applications are made to multireference correlation in the potential energy surfaces of the molecules $\mathrm{HF}$, ${\mathrm{H}}_{2}\mathrm{O}$, and ${\mathrm{C}}_{2}$ in polarized valence double-zeta basis sets while ${\mathrm{N}}_{2}$ is treated in polarized valence double- and triple-zeta basis sets. Accurate ground-state energies and 1-RDM occupation numbers are obtained at both equilibrium and nonequilibrium geometries where the energies are within a few millihartrees of those from full configuration interaction.

Correlated Linear Response Calculations of the C6 Dispersion Coefficients of Hydrogen Halides

Three correlated linear response theory methods - the second order polarization propagator approximation (SOPPA), the second order polarization propagator approxima- tion with coupled cluster singles and doubles amplitudes, SOPPA(CCSD), and multiconfig- urational self-consistent field (MCSCF) linear response theory - were used to determine the electric dipole polarizabilities at imaginary frequencies of the hydrogen halides HX (with X = F, Cl, Br and I). The dynamic polarizabilities were subsequently used to calculate the C6 dispersion coefficient for a pair of interacting HX molecules via numerical integra- tion of the Casimir-Polder formula. The dependence of the polarizabilities, their frequency dependence and C6 coefficients on the one-electron basis set, the level of correlation and zero-point vibrational corrections was studied. Furthermore it was investigated whether relativistic effective core potentials (RECP) could be employed in such calculations. It was found that SOPPA(CCSD) and MCSCF calculations with large active spaces using basis sets of at least double augmented quadruple zeta quality give comparable results which in particular for SOPPA(CCSD) are in close agreement with the available experimental values. Furthermore it could be shown that the results of the RECP calculations are in close agreement with the all electron calculations.

One-Dimensional Zigzag Chains of Cs-: The Structures and Properties of Li+(Cryptand[2.1.1])Cs- and Cs+(Cryptand[2.2.2])Cs-

The crystal structure and properties of lithium (cryptand[2.1.1]) ceside, Li+ (C211)Cs-, are reported. Li+ (C211)Cs- is the second ceside and third alkalide with a one-dimensional (1D) zigzag chain of alkali metal anions. The distance between adjacent Cs- anions, 6 A, is shorter than the sum of the van der Waals radii, 7 A. Optical, magic angle spinning NMR, two-probe alternating and direct current conductivity, and electron paramagnetic resonance measurements reveal unique physical properties that result from the overlap of adjacent Cs- wave functions in the chain structure. The properties of cesium (cryptand[2.2.2]) ceside, Cs+ (C222)Cs-, were also studied to compare the effects of the subtle geometric changes between the two 1D zigzag chain structures. Li+ (C211)Cs- and Cs+ (C222)Cs- are both low-band-gap semiconductors with anisotropic reflectivities and large paramagnetic 133Cs NMR chemical shifts relative to Cs- (g). An electronic structure model consistent with the experimental data has sp2-hybridized Cs- within the chain and sp-hybridized chain ends. Ab initio multiconfiguration self-consistent field calculations on the ceside trimer, Cs3(3-), support this model and indicate a net bonding interaction between nearest neighbors. The buildup of electron density between adjacent Cs- anions is visualized through an electron density difference map constructed by subtracting the density of three cesium atoms from the short Cs3(3-) fragment.

Stability and Thermal Rearrangement of (E,E)-1,3-Cycloheptadiene and trans-Bicyclo[3.2.0]hept-6-ene

The highly strained (E,E)-1,3-cycloheptadiene was shown to be a minimum on the potential energy surface; two structural isomers were found at the MP2 level, but multiconfiguration self-consistent field calculations show that only one is a true minimum. The isomerization of (E,E)-1,3-cycloheptadiene was investigated through double bond rotation, and electrocyclic ring closure. The first pathway gives (E,Z)-1,3-cycloheptadiene, with a barrier of 7.2 kcal x mol(-1), and the second pathway gives the trans isomer of bicyclo[3.2.0]hept-6-ene with a barrier of 13.0 kcal x mol(-1). The strain energy of (E,E)-1,3-cycloheptadiene was calculated using homodesmotic reactions and found to be about 96 kcal x mol(-1) whereas that for (E,Z)-1,3-cycloheptadiene was only 38 kcal x mol(-1), implying that the second trans double bond imparts an additional 58 kcal x mol(-1) in strain energy. The trans isomer of bicyclo[3.2.0]hept-6-ene was calculated to have a strain energy of 69 kcal x mol(-1) and a barrier of 27 kcal x mol(-1) for isomerization to (Z,Z)-1,3-cycloheptadiene. Although many of the structures reported here could be described using a single determinant wave function, several could not, making a multireference method necessary for a complete description of the potential energy surface.

Multiconfiguration self-consistent-field theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the multiconfiguration self-consistent-field (MCSCF) theory. One- and two-layer approaches were developed, the former involving all dimer MCSCF calculations and the latter limiting MCSCF calculations to a small part of the system. The accuracy of the two methods was tested using the six electrons in six orbitals complete active space type of MCSCF and singlet spin state for phenol+(H(2)O)(n), n=16,32,64 (6-31G( *) and 6-311G( *) basis sets); alpha helices and beta strands of phenylalanine-(alanine)(n), n=4,8,16 (6-31G( *)). Both double-zeta and triple-zeta quality basis sets with polarization were found to have very similar accuracy. The error in the correlation energy was at most 0.000 88 a.u., the error in the gradient of the correlation energy was at most 6.x10(-5) a.u./bohr and the error in the correlation correction to the dipole moment was at most 0.018 D. In addition, vertical singlet-triplet electron excitation energies were computed for phenol+(H(2)O)(n), (n=16,32,64), 6-31G( *), and the errors were found to be at most 0.02 eV. Approximately linear scaling was observed for the FMO-based MCSCF methods. As an example, an FMO-based MCSCF calculation with 1262 basis functions took 98 min on one 3.0 GHz Pentium4 node with 1 Gbyte RAM.

Calculation of nonadiabatic couplings in density-functional theory

This paper proposes methods for calculating the derivative couplings between adiabatic states in density-functional theory (DFT) and compares them with each other and with multiconfigurational self-consistent field calculations. They are shown to be accurate and, as expected, the costs of their calculation scale more favorably with system size than post-Hartree-Fock calculations. The proposed methods are based on single-particle excitations and the associated Slater transition-state densities to overcome the problem of the unavailability of multielectron states in DFT which precludes a straightforward calculation of the matrix elements of the nuclear gradient operator. An iterative scheme employing linear-response theory was found to offer the best trade-off between accuracy and efficiency. The algorithms presented here have been implemented for doublet-doublet excitations within a plane-wave-basis and pseudopotential framework but are easily generalizable to other excitations and basis sets. Owing to their fundamental importance in cases where the Born-Oppenheimer separation of motions is not valid, these derivative couplings can facilitate, for example, the treatment of nonadiabatic charge transfers, of electron-phonon couplings, and of radiationless electronic transitions in DFT.

The “JK-only” approximation in density matrix functional and wave function theory

Various energy functionals applying the "JK-only" approximation which leads to two-index two-electron integrals instead of four-index two-electron integrals in the electron-electron interaction term of the electronic energy are presented. Numerical results of multiconfiguration self-consistent field calculations for the best possible "JK-only" wave function are compared to those obtained from the pair excitation multiconfiguration self-consistent (PEMCSCF) method and two versions of density matrix functional theory. One of these is derived making explicit use of some necessary conditions for N representability of the second-order density matrix. It is shown that this method models the energy functional based on the best possible "JK-only" wave function with good accuracy. The calculations also indicate that only a minor fraction of the total correlation energy is incorporated by "JK-only" approaches for larger molecules.

Accurate calculation of core-electron binding energies: multireference perturbation treatment.

Multireference perturbation theory (MRPT) with multiconfigurational self-consistent field (MCSCF) reference functions is applied to the calculations of core-electron binding energies (CEBEs) of atoms and molecules. Orbital relaxations in a core-ionized state and electron correlation are both taken into account in a conventional MCSCF-MRPT procedure. In the MCSCF calculation, the target core ionized state is directly optimized as an excited state and this treatment can completely prevent a variational collapse. Multireference Moller-Plesset perturbation theory and multiconfigurational self-consistent field reference quasidegenerated perturbation theory were used to treat electron correlation. The present method quite accurately reproduced the 1s CEBEs of CH4, NH3, H2O, and FH; the average deviation from the experimental data is 0.11 eV using Ahlrichs' VTZ basis set. The C 1s and O 1s CEBEs of formic acid and acetic acid were calculated and the results are consistent with the bonding characters of the atoms in these molecules. The present procedure can also be applied to CEBEs of higher angular momentum orbitals by including spin-orbit coupling. The calculated CEBEs of Ar 2p, HCl 2p, Kr 3d, and HBr 3d are in reasonable agreement with the available experimental values. In the calculation of the 3d CEBEs, a relativistic correction significantly improves the agreements. The effect of polarization functions is also discussed.

Multiconfigurational Self-Consistent Field Study of the Excited State Properties of 4-Thiouracil in the Gas Phase

A comprehensive theoretical study of the vertical singlet and triplet electronic transitions of 4-thiouracil was performed at the multiconfigurational self-consistent field (MCSCF) level of theory. The ground state geometry of the molecule was optimized at the MP2/6-311++G(d,p) level. The MCSCF calculations were performed using the 6-311+G(d) basis set. The active space for the MCSCF calculations consisted of 12 orbitals in which 6 orbitals were the occupied π type while the remaining 6 orbitals were the virtual π* type. To compute the nπ* transitions, the two occupied orbitals were replaced with two σ orbitals localized at the thiocarbonyl and carbonyl groups, respectively. Further, MCSCF calculations were also performed with a slightly smaller active space where 10 electrons were distributed in 11 orbitals. The effects of dynamic electron correlation on the MCSCF energies were considered at the second-order multiconfigurational quasi-degenerate perturbation (MCQDPT2) theory. The computed transition energies after the MCQDPT2 correlation were found to be in agreement with the experimental data. The ground and excited state molecular electrostatic potentials and the Mulliken charge distributions in different states were also investigated.

Theoretical study of the reaction of H2 with a Cu2Pt2 cluster

The study of the interaction of a pyramidal tetramer of Cu2Pt2 with the H2 is reported here through ab initio multiconfigurational self-consistent field (MC-SCF) calculations, plus extensive multireference configuration interaction (MR-CI), variational and perturbative calculations. The lowest three electronic states X 1A′, a 3A′ and a 1A′ of the bare cluster were considered in order to study this interaction. For the H2 Cs approaching a Pt vertex, results show that the Cu2Pt2 pyramid cluster in its X 1A′ and a 1A′ states can spontaneously capture and dissociate the H2. For the H2 Cs approaching a Cu vertex, where H2 is located in the Cs reflecting plane, the Cu2Pt2 cluster in its X 1A′ electronic state shows capture of the hydrogen molecule after surmounting an energy barrier; moreover, in this approach the Cu2Pt2 cluster in its a 1A′ electronic state shows spontaneous capture of the hydrogen molecule. For the H2 approaching a Cu vertex, where the Cs reflecting plane bisects the H2 molecule, the Cu2Pt2 cluster in its three lowest-lying states is able to capture the hydrogen molecule after surmounting a small barrier. The Cu2Pt2+H2 Cs face-on interactions show a lower H2 activation than that which was obtained in the equivalent Pt4+H2 interactions.