Molecular applications of the intermediate Hamiltonian Fock-space coupled-cluster method for calculation of excitation energies

The intermediate Hamiltonian Fock-space coupled-cluster (FS-CC) method with singles and doubles is applied to calculate vertical excitation energies (EEs) for some molecular systems. The calculations are performed for several small molecules, such as H2O, N2, and CO, and for larger systems, such as C2H4, C4H6, and C6H6. Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FS-CC equations, and the efficient factorization strategy, relatively large basis sets and model spaces are employed permitting a comparison of the calculated vertical EEs with the experimental data.     

Multireference Fock space coupled cluster method in the effective and intermediate Hamiltonian formulation for the (2,0) sector

The effective and intermediate Hamiltonian (IH) multireference coupled cluster (CC) method with singles (S) and doubles (D) within the double electron attached (2,0) sector of the Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (2,0) FS problem allows to replace the iterative scheme of the FS-CC equations based on the effective Hamiltonian with the diagonalization of the properly constructed matrix. The proposed method, IH-FS-CCSD (2,0), is rigorously size-extensive, easy to code, and numerically very efficient with the results comparable or slightly better than equation-of-motion ones at the CCSDT (T--triples) level. The performance of the method is discussed on the basis of test calculations for potential energy curves of the systems for which double positive ions dissociate into closed shell fragments (e.g., Na(2) dimer). The double electron attachment (DEA) scheme can be also useful in determination of the excitation spectra for difficult cases. The example is a carbon atom which has two electrons out of the closed shell structure. The newly implemented method is also analyzed by plotting potential energy curve for twisted ethylene case as a function of a dihedral angle between two methylene groups. Using DEA scheme one obtains a smooth, cusp free curve.     

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     

Calculation of 13C-NMR chemical shift using the intermediate neglect of differential overlap model

The local origin/local orbital (LORG ) method of Hansen and Bouman has been implemented with the intermediate neglect of differential overlap Hamiltonian for spectroscopy (INDO /S ). The method is shown capable of demonstrating the inductive effects associated with electron-withdrawing substituents through the diamagnetic shielding term. In addition, the method is capable of differentiating chemical shift in differing bond environments. The calculated paramagnetic contribution, however, is deficient for substituents that saturate the minimal basis such as oxygen and fluorine, which severely limits the general utility of the procedure. Through the utilization of reduced linear equations for the paramagnetic term, the method is amenable to any molecule for which a self-consistent field can be performed and therefore can potentially be used to study very large systems. At present, however, the LORG method when used with the rapid INDO /S model Hamiltonian does not reliably reproduce the paramagnetic contribution to the shielding.     

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for correctly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian (J=0) in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OClO molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.     

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.     

Accurate relativistic energy-consistent pseudopotentials for the superheavy elements 111 to 118 including quantum electrodynamic effects

Energy-consistent two-component semi-local pseudopotentials for the superheavy elements with atomic numbers 111-118 have been adjusted to fully relativistic multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian, including perturbative corrections for the frequency-dependent Breit interaction in the Coulomb gauge and lowest-order quantum electrodynamic effects. The pseudopotential core includes 92 electrons corresponding to the configuration [Xe]4f(14)5d(10)5f(14). The parameters for the elements 111-118 were fitted by two-component multi-configuration Hartree-Fock calculations in the intermediate coupling scheme to the total energies of 267 up to 797 J levels arising from 31 up to 62 nonrelativistic configurations, including also anionic and highly ionized states, with mean absolute errors clearly below 0.02 eV for averages corresponding to nonrelativistic configurations. Primitive basis sets for one- and two-component pseudopotential calculations have been optimized for the ground and excited states and exhibit finite basis set errors with respect to the finite-difference Hartree-Fock limit below 0.01 and 0.02 eV, respectively. General contraction schemes have been applied to obtain valence basis sets of polarized valence double- to quadruple-zeta quality. Results of atomic test calculations in the intermediate coupling scheme at the Fock-space coupled-cluster level are in good agreement with those of corresponding fully relativistic all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian. The results demonstrate besides the well-known need of a relativistic treatment at the Dirac-Coulomb level also the necessity to include higher-order corrections for the superheavy elements.     

Visualizing the zero order basis of the spectroscopic Hamiltonian

Recent works have shown that a generalization of the spectroscopic effective Hamiltonian can describe spectra in surprising regions, such as isomerization barriers. In this work, we seek to explain why the effective Hamiltonian is successful where there was reason to doubt that it would work at all. All spectroscopic Hamiltonians have an underlying abstract zero-order basis (ZOB) which is the "ideal" basis for a given form and parameterization of the Hamiltonian. Without a physical model there is no way to transform this abstract basis into a coordinate representation. To this end, we present a method of obtaining the coordinate space representation of the abstract ZOB of a spectroscopic effective Hamiltonian. This method works equally well for generalized effective Hamiltonians that encompass above-barrier multiwell behavior, and standard effective Hamiltonians for the vicinity of a single potential minimum. Our approach relies on a set of converged eigenfunctions obtained from a variational calculation on a potential surface. By making a one-to-one correspondence between the energy eigenstates of the effective Hamiltonian and those of the coordinate space Hamiltonian, a physical representation of the abstract ZOB is calculated. We find that the ZOB basis naturally adjusts its complexity depending on the underlying nature of phase space, which allows spectroscopic Hamiltonians to succeed for systems sampling multiple stationary points.     

A model for the Heyrovsky reaction as the second step in hydrogen evolution

On a number of electrodes the second step in hydrogen evolution is the reaction of a proton with an adsorbed hydrogen intermediate to form a molecule, which is also known as the Heyrovsky reaction. We have developed a model Hamiltonian for this reaction, which for concrete applications requires extensive calculations on the basis of density-functional theory. Explicit results are presented for a Ag(111) electrode. The rate-determining step is electron transfer to the proton that approaches the electrode from the solution. At the saddle point for this reaction the adsorbed hydrogen atom has moved a little away from the surface in order to reduce the repulsion of the product molecule. Electron transfer to the proton occurs when the distance between the two particles is close to the bond distance of the hydrogen molecule.     

On the quantum field theory of photoionisation and electron scattering reactions on atoms

Brillouin-Wigner perturbation theory, formulated in the Schrödinger picture of quantum field theory, is employed to derive a perturbative scheme for the scattering matrix for photoionisation and electron scattering reactions on atoms. It is important to note that the intermediate states appearing in this series are physical, i.e. fully correlated, eigenstates of the total Hamiltonian. The scheme is amenable to numerical analysis: The key point is the use of an “energy-optimised”g-Hartree basis which yields an efficient treatment of atomic correlations and makes Brillouin-Wigner and Rayleigh-Schrödinger perturbation theories coincide.     

Canonical transcorrelated theory with projected Slater-type geminals

An effective Hamiltonian perturbed with explicit interelectronic correlation is derived from similarity transformation of Hamiltonian using a unitary operator with Slater-type geminals. The Slater-type geminal is projected onto the excitation (and deexcitation) component as in the F12 theory. Simplification is made by truncating higher-body operators, resulting in a correlated Hamiltonian which is Hermitian and has exactly the same complexity as the original Hamiltonian in the second quantized form. It can thus be easily combined with arbitrary correlation models proposed to date. The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite. Our theory is naturally extensible to multireference calculations on the basis of the generalized normal ordering. The construction of the effective Hamiltonian is non-iterative. The numerical assessments demonstrate that the present scheme improves the basis set convergence of the post-mean-field calculations at a similar rate to the explicitly correlated methods proposed by others that couple geminals and conventional excitations.     

琥乙红霉素在碳糊电极上的电化学测定方法

以碳糊电极为工作电极,采用循环伏安（CV）法和示差脉冲伏安（DPV）法研究了琥乙红霉素（EEs）在电极上的电化学行为,建立了一种测定EEs的电化学新方法。 研究结果表明,在0.1 mol/L磷酸盐（pH=8.0）的缓冲液中,EEs在0.83和0.97 V(vs.SCE）处出现2个氧化峰。 用计时安培法(I-t)对EEs进行定量分析,峰电流Ip与琥乙红霉素的浓度分别在2.0×10-7~2.8×10-6 mol/L和2.8×10-6~3.1×10-5 mol/L范围内呈良好的线性关系检出限（S/N=3）为1.0×10-7 mol/L。 采用标准加入法测得回收率为950%～988%,RSD为3.4%(n=3)。 该方法具有较高的选择性和灵敏度,可用于药剂中EEs含量的测定,结果令人满意。     

Multi-reference Fock space coupled-cluster method in the intermediate Hamiltonian formulation for potential energy surfaces

The effective and intermediate Hamiltonian multi-reference coupled-cluster (CC) method with singles and doubles for the doubly ionized (0,2) sector of Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (0,2) FS problem provides a robust computational scheme for solving the FS-CC equations free from the intruder state problem. By introducing an efficient factorization strategy, we obtain a very efficient tool that can be used for computing double ionization potentials but more significantly to describe multi-reference problems in CC theory, illustrated by twisted ethylene and the potential energy curve for F(2). The latter separates smoothly to two F atoms, while the former avoids the cusp behavior at the 90° dihedral. We also explore the double ionization potentials for several small molecules, H(2)O, CO, C(2)H(2), and C(2)H(4).     

A simulation of the photoelectron spectrum of pyrazolide

Building on previous theoretical and spectroscopic studies of the pyrazolyl radical, a new three-state quasidiabatic Hamiltonian is reported which reproduces not only the equilibrium geometries and harmonic frequencies of the nominal X (2)A(2) state and low-lying A (2)B(1) excited state, but also the minimum energy points on the lowest two-state (X (2)A(2), A (2)B(1)) and three-state (X (2)A(2), A (2)B(1), B (2)B(2)) seams of conical intersection. The three-state Hamiltonian includes all terms through second order in both the diagonal and off-diagonal blocks. Its construction is accomplished in two steps. First, a nascent Hamiltonian, centered at the lowest energy two-state conical intersection, is determined using ab initio gradients and derivative couplings. Then, the nascent Hamiltonian is improved by optimizing selected contributions to the second-order coefficients to better reproduce relevant minima and harmonic frequencies. This Hamiltonian is then expressed in a basis tailored to describe the neutral states of interest under the multimode vibronic coupling approximation. The vibronic Hamiltonian is diagonalized to obtain negative ion photoelectron spectra for pyrazolide-h(3) and the completely deuterated analog pyrazolide-d(3). The resultant spectra, determined employing vibronic Hamiltonians as large as 500 million terms, compare favorably to recent theoretical and spectroscopic results for pyrazolyl-d(3) and to spectroscopic results for pyrazolyl-h(3), for which no reliable simulations had been available.     

Medium and interfacial effects in the multistep reduction of binuclear complexes with robson-type ligand

We present a combined experimental and computational approach to the modeling and prediction of reactivity in multistep processes of heterogeneous electron transfer. The approach is illustrated by the study of a Robson-type binuclear complex (-Cu(II)-Cu(II)-) undergoing four-electron reduction in aqueous media and water-acetonitrile mixtures. The observed effects of solvent, pH, buffer capacity, and supporting electrolyte are discussed in the framework of a general reaction scheme involving two main routes; one of them includes protonation of intermediate species. The main three problems are addressed on the basis of modern charge transfer theory: (1) the effect of the nature of reactant and intermediate species (protonated/deprotonated, bare or associated with supporting anion/solvent molecule) on the standard redox potential, the electronic transmission coefficient, and the intramolecular reorganization; (2) possible effect of protonation on the shape of the reaction free energy surfaces which are built using the Anderson Hamiltonian; (3) electron transfer across an adsorbed chloride anion. Quantum chemical calculations were performed at the density functional theory level.     

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

Eigenvalue lower bounds with Bazley’s special choice of an infinite-dimensional subspace

Bazley’s special choice of a finite-dimensional space to construct an intermediate operator between a base operator and the full Hamiltonian is a standard technique to calculate lower bounds to the energies of a system. We modify Bazley’s method to accommodate an infinite-dimensional space that is complete in one particle of the system. An application to the helium atom shows improvement in the lower bound to the ground-state energy, indicating promise in our method. However, significant problems are revealed which include (1) poorer bounds for the excited states, (2) lack of symmetry in the intermediate operator, and (3) lack of direction for improvement.     

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: energy levels of coupled systems with as many as 16 coordinates

In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.