Convergence property of multireference many-body perturbation theory analyzed by the use of a norm of effective Hamiltonian

Summary It is proposed to use a norm of anth order effective Hamiltonian, for analyzing the convergence property of the multireference many-body perturbation theory (MR-MBPT). The utilization of the norm allows us to employ only (1) asingle number for all the states that we are interested in, and (2) values which decreases from thepositive side to zero as the ordern of the perturbation increases. This characteristic features are in contrast to those in the usually used scheme whereseveral numbers, namely, the eigenvalues of the target states, should be used and they mayoscillate around exact eigenvalues. The present method has been applied to MR-MBPT calculations of the (H₂)₂, CH₂, and LiH molecules based on the multireference versions of Rayleigh-Schrödinger PT, Kirtman-Certain-Hirschfelder PT, and the canonical Van Vleck PT; and following features are found: (1) the above three versions of the perturbation theories have essentially the same convergence property judged from the lowering of the norm; (2) the lower order truncation of the perturbation series gives reasonable solutions; (3) the norm decreases irrespective of the perturbation expansion being convergent or divergent for the first several orders (up to about the sixth order).     

Superior performance of Mukherjee’s state-specific multi-reference coupled-cluster theory at the singles and doubles truncation scheme with localized active orbitals

The state-specific multi-reference coupled-cluster (SS-MRCC) theory of Mukherjee et al., in its singles and doubles truncation scheme (SS-MRCCSD), misses important couplings between the virtual functions reached by single and double excitations from different model functions. Since the SS-MRCC theory is not invariant with respect to the transformations among the active orbitals, the results are dependent on the active orbitals chosen. We demonstrate in this paper with results for potential energy curves for several example molecules involving single and multiple bond dissociation that the performance of SS-MRCCSD is significantly improved if localized active orbitals are used. The improvement is remarkable both in terms of the non-parallelity error and the magnitude of correlation energy recovered vis-a-vis the full configuration interaction results with the same basis set. The results bolster our claim that SS-MRCCSD with localized orbitals is an accurate general theory for potential energy surfaces.     

An application of second-order n-electron valence state perturbation theory to the calculation of excited states

n–electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory where all the zero-order wave functions are of multireference nature, being generated as eigenfunctions of a two–electron model Hamiltonian. The absence of intruder states makes NEVPT an interesting choice for the calculation of electronically excited states. Test calculations have been performed on several valence and Rydberg transitions for the formaldehyde and acetone molecules; the results are in good accordance with the best calculations and with the existing experimental data.Contribution to the Jacopo Tomasi Honorary Issue     

Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies

In this paper we propose and numerically implement a specific scheme for calculating the excitation energies (EEs) within the Fock space multireference coupled cluster framework, which includes the contributions from noniterative triples cluster amplitudes. These contribute to the EEs at the third order. We present results for CH⁺ and N2, and study the effects of these noniterative triples on EEs. Received: 28 July 1997 / Accepted: 8 December 1997     

Application of the coupled-cluster theory to atomic frequency-dependent polarizabilities

Summary A time-dependent coupled-cluster approach may be employed to describe dynamic processes of many-electron systems. Atomic properties, such as the frequency-dependent polarizability, can be treated as a response of the system described by the coupled-cluster expansion to an external radiation field. The major difficulty in the realization of such a formalism is to deal with dynamic pair functions. The procedure reported here is to simplify the full set of single- and pair-excitation expansion equations to a subset of equations which includes polarization and relaxation effects to all orders and is solved by using a complete set of discrete basis functions. Calculations of excitation energies and frequency-dependent electric dipole polarizabilities for helium are presented. Application of the procedure to calculate photoionization cross sections is discussed.     

A multireference configuration interaction study with singles and doubles of some mesoionic rings: reaction and activation free energies for the ring-opening reaction

For the first time, multireference configuration interaction with singles and doubles (MR-CISD) calculations (including extensivity corrections, MR-CISD+Q) have been performed to study the ring-opening reactions of the following mesoionic rings and their 2, 3, and 4 methyl-substituted derivatives: ( a ) 1,3-oxazol-5-one, ( c ) 1,3-oxazol-5-thiolate, and ( d ) 1,3-thiazol-5-thiolate. The ring-opening reaction of the parent 1,3-thiazol-5-one mesoionic ring ( b ) has also been studied. The effect of methyl and S replacement on the reaction and activation free energies (ΔG and ΔG^†, respectively) is studied. For a , the effect of methyl replacement on C2 is almost negligible, while on N3 and C4, it is significantly larger, especially in the latter position. The open structure (ketene form) of a is considerably more stable, and the replacement of the exocyclic O by S stabilizes the ketene tautomer even more and increases ΔG^† considerably. On the other hand, replacement of the endocyclic O by S (yielding b ) completely prevents the formation of the open structure, while replacement of both endo and exo atoms strongly stabilizes the cyclic structure and leads to a remarkably high value of ΔG^†. In some cases, the relative polarity of the stationary points is used to estimate how ΔG and ΔG^† are expected to change as one goes from the gas phase to aprotic polar solvents. A linear correlation between the multiconfigurational character of the mesoionic rings and ΔG^† has been observed, and such correlation is used to estimate the ΔG^† values for the methyl-substituted c and d . The largest barrier (13.78 kcal/mol) has been obtained for the 4-methyl-1,3-thiazol-5-thiolate (4-m- d ).     

Intruder states in multireference perturbation theory: the ground state of manganese dimer

A detailed analysis of a severe intruder state problem in the multistate multireference perturbation theory (MS-MRPT) calculations on the ground state of manganese dimer is presented. An enormous number of detected intruder states (> 5000) do not permit finding even an approximate shape of the X(1)Sigma(g) (+) potential energy curve. The intruder states are explicitly demonstrated to originate from quasidegeneracies in the zeroth-order Hamiltonian spectrum. The electronic configurations responsible for appearance of the quasidegeneracies are identified as single and double excitations from the active orbitals to the external orbitals. It is shown that the quasidegeneracy problem can be completely eliminated using shift techniques despite of its severity. The resultant curves are smooth and continuous. Unfortunately, strong dependence of the spectroscopic parameters of the X(1)Sigma(g) (+) state on the shift parameter is observed. This finding rises serious controversies regarding validity of employing shift techniques for solving the intruder state problem in MS-MRPT. Various alternative approaches of removing intruder states (e.g., modification of the basis set or changing the active space) are tested. None of these conventional techniques is able to fully avoid the quasidegeneracies. We believe that the MS-MRPT calculations on the three lowest A(g) states of manganese dimer constitute a perfect benchmark case for studying the behavior of MRPT in extreme situations.     

A truncated version of reduced multireference coupled-cluster method with singles and doubles and noniterative triples: application to F2 and Ni(CO)n (n=1, 2, and 4)

A perturbatively truncated version of the reduced multireference coupled-cluster method with singles and doubles and noniterative triples RMR CCSD(T) is described. In the standard RMR CCSD method, the effect of all triples and quadruples that are singles or doubles relative to references spanning a chosen multireference (MR) model space is accounted for via the external corrections based on the MR CISD wave function. In the full version of RMR CCSD(T), the remaining triples are then handled via perturbative corrections as in the standard, single-reference (SR) CCSD(T) method. By using a perturbative threshold in the selection of MR CISD configuration space, we arrive at the truncated version of RMR CCSD(T), in which the dimension of the MR CISD problem is significantly reduced, thus leaving more triples to be treated perturbatively. This significantly reduces the computational cost. We illustrate this approach on the F2 molecule, in which case the computational cost of the truncated version of RMR CCSD(T) is only about 10%-20% higher than that of the standard CCSD(T), while still eliminating the failure of CCSD(T) in the bond breaking region of geometries. To demonstrate the capabilities of the method, we have also used it to examine the structure and binding energy of transition metal complexes Ni(CO)n with n=1, 2, and 4. In particular, Ni(CO)2 is shown to be bent rather than linear, as implied by some earlier studies. The RMR CCSD(T) binding energy differs from the SR CCSD(T) one by 1-2 kcal/mol, while the energy barrier separating the linear and bent structures of Ni(CO)2 is smaller than 1 kcal/mol.     

Extrapolated intermediate Hamiltonian coupled-cluster approach: theory and pilot application to electron affinities of alkali atoms

The intermediate Hamiltonian (IH) coupled-cluster method makes possible the use of very large model spaces in coupled-cluster calculations without running into intruder states. This is achieved at the cost of approximating some of the IH matrix elements, which are not taken at their rigorous effective Hamiltonian (EH) value. The extrapolated intermediate Hamiltonian (XIH) approach proposed here uses a parametrized IH and extrapolates it to the full EH, with model spaces larger by several orders of magnitude than those possible in EH coupled-cluster methods. The flexibility and resistance to intruders of the IH approach are thus combined with the accuracy of full EH. Various extrapolation schemes are described. A pilot application to the electron affinities (EAs) of alkali atoms is presented, where converged EH results are obtained by XIH for model spaces of approximately 20,000 determinants; direct EH calculations converge only for a one-dimensional model space. Including quantum electrodynamic effects, the average XIH error for the EAs is 0.6 meV and the largest error is 1.6 meV. A new reference estimate for the EA of Fr is proposed at 486+/-2 meV.     

Electronic structure and optical spectra of transition metal complexes by the effective Hamiltonian method

Summary A semiempirical effective Hamiltonian treatment is proposed for transition metal complexes, taking into accountd-electron correlations, weak covalency of the metal-ligand bonds and the electronic structure of the ligand sphere. The technique uses the variation wave function which differs from the usual Hartree-Fock antisymmetrized product of molecular orbitals extended over the whole complex. The scheme is implemented and parameters describing the metal-ligand interactions are adjusted to reproduced-d-excitation spectra of a number of octahedral MF ₆ ^4– (M=Mn, Fe, Co, Ni) anions, Mn(FH) ₆ ²⁺ cation, CoCl ₆ ^4– anion, and a tetrahedral CoCl ₄ ^2– anion. The values of the parameters are reasonable, thus confirming the validity of the proposed scheme.     

Ab initio computation of the interaction energy curves for some low-lying states of CO which dissociate to ground-state3 P O and3 P C atoms

Summary We investigate the molecular electronic structure of the quintet states of CO which correspond to the C(³ P)+O(³ P) interaction at several levels of theory. We find the 1⁵⁺ state to be relatively deeply bound (D _e ca. 587 cm^–1) while the other quintets have relatively shallow potential wells (D _e<40 cm^–1) according to our multireference configuration interaction calculations which are counterpoise corrected for basis set superposition effects. Our results are in qualitative accord with the recent semiempirical estimates of Bussery and co-workers [(1989) Chem. Phys. 134:7].National Academy of Sciences, National Research Council, Air Force Astronautics Laboratory, Resident Research Associate 1987–89     

Analytic UHF-CCSD(T) second derivatives: implementation and application to the calculation of the vibration-rotation interaction constants of NCO and NCS

An implementation of analytic open-shell UHF-CCSD(T) second derivatives is presented. To demonstrate applicability and test the accuracy of the UHF-CCSD(T) approach for the determination of spectroscopical parameters, vibration-rotation interaction constants were calculated for the ground (1²Π) and first electronically excited (1²Σ) states of the NCO and NCS radicals. In addition, harmonic vibrational frequencies for both states, the Renner-Teller parameter for the ground state, as well as the 1²Π→1²Σ excitation energy are reported. While the computed values are in good agreement with reliable experimental information for NCO, most of the data presented for NCS are predictions of quantities not well known from experiment. Received: 4 May 1998 / Accepted: 27 July 1998 / Published online: 12 October 1998     

Degenerate symmetry-adapted perturbation theory of weak interactions between closed- and open-shell monomers: application to Rydberg states of helium hydride

Symmetry-adapted perturbation theory is extended to the (quasi) degenerate, open-shell case. The new formalism is tested in calculations of the interaction energies for a helium atom in the ground state interacting with an excited hydrogen atom. It is shown that the method gives satisfactory results if the coupling with higher Rydberg states of the dimer is small, as is the case for the A²Σ⁺,B²Π,E²Π,3²Π, and 1²Δ states of HeH. For the C²Σ⁺ state convergence of the method is very slow, but it can be improved by including the n=3 states in the model space. Received: 3 June 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

A simple correction to final state energies of doublet radicals described by equation-of-motion coupled cluster theory in the singles and doubles approximation

A computationally inexpensive energy correction is suggested for radicals described by the equation-of-motion coupled cluster method for ionized states in the singles and doubles approximation (EOMIP-CCSD). The approach is primarily intended for doublet states that are qualitatively described by Koopmans' approximation. Following a strategy similar to those used in multireference coupled cluster theory, the proposed correction accounts for all correlation effects through third order in perturbation theory and also includes selected contributions to higher-order energies. As an initial test of the numerical performance of the method, total energies and energy splittings are calculated for some small prototype radicals.     

A challenge for density functional theory: the XONO and XNO2 (X=F, Cl, and Br) molecules

The equilibrium geometries, harmonic frequencies, dipole moments, infrared intensities, and relative energies of the cis-XONO, trans-XONO, and XNO₂ (X=F, Cl, and Br) have been investigated using four functionals in common use in Kohn-Sham density functional theory (DFT) calculations. Two of the functionals include non-local or gradient correction terms, while the other two also incorporate some exact Hartree-Fock exchange and are labeled hybrid functionals. The quality of the results obtained from the functionals is determined by comparison to previously published high-level coupled-cluster calculations. The hybrid functionals perform better for prediction of the equilibrium geometries, where the two gradient corrected functionals yield qualitatively incorrect molecular structures for cis-FONO and cis-ClONO. None of the functionals perform well in predicting all six harmonic frequencies, showing that the correlation between equilibrium geometries and harmonic frequencies is not as strong for these DFT methods as it is for conventional wavefunction ab initio methods, such as coupled-cluster theory. Results from the various functionals generally come into better agreement with each other and also with the coupled-cluster results moving down the periodic table. Received: 12 February 1997 / Accepted: 25 March 1997     

Excited States of Conjugated Hydrocarbons Using the Molecular Mechanics - Valence Bond (MMVB) Method: Conical Intersections and Dynamics

We developed the molecular mechanics—valence bond (MMVB) method following an original suggestion of Jean-Paul Malrieu and coworkers. By coupling a parameterized Heisenberg Hamiltonian to a standard classical force field (MM2), reliable ground and excited state geometries of conjugated hydrocarbons can be rapidly optimized. The MMVB method was central to our development of algorithms for locating conical intersections and calculating their associated decay dynamics. Here, we briefly review the chemical applications of MMVB to date, and present two new studies using the photostability of pyracylene and the excited state decay dynamics of the photochromic dihydroazulene/vinylheptafulvene (DHA/VHF) reaction.     

Transition states in modern valence-bond theory: application to the Cope rearrangement

Modern valence-bond theory, in its spin-coupled form, is used to study the electronic structure of the transition structures in the Cope rearrangement. It is found that the transition structure described by a “chair” geometry with a “6-in-6” CASSCF/6-31G^* wave function is clearly aromatic while the CASSCF/6-31G^*“boat” transition structure corresponds more closely to two weakly interacting allyl radicals. Moreover, there is a striking resemblance between the CASSCF chair transition structure and the benzene molecule, arising from the modern valence-bond analysis in terms of Rumer spin functions. In agreement with previous works, dynamical correlated wave functions show shorter interallylic distances in the optimized transitions structures. The use of spin-coupled wave functions on the latter geometries results in diradical and aromatic character for the chair and boat transition structures, respectively. Received: 13 October 1998 / Accepted: 30 December 1998 / Published online: 7 June 1999     

An integrated computational tool for the study of the optical properties of nanoscale devices: application to solar cells and molecular wires

We present a combined computational strategy for the study of the optical properties of nanoscale systems, using a combination of codes and techniques based on Density Functional Theory (DFT) and its Time Dependent extension (TDDFT). In particular, we describe the use of Car–Parrinello molecular dynamics simulations for the study of nanoscale devices and show the integration of the obtained results with available quantum chemistry codes for the calculation of TDDFT excitation energies, including solvation effects by continuum solvation models. We review some prototypical applications of this integrated computational strategy, ranging from the interaction of dye sensitizers with TiO₂ nanoparticles, of interest in the field of dye-sensitized solar cells, to transition metal molecular wires exceeding 3 nm length.