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.     

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.     

The expansion of hydrogen states in Gaussian orbitals

The convergence properties of Gaussian orbitals are studied by considering a very simple system, the hydrogen atom. We have variationally optimized even-tempered basis sets containing up to 60 s functions for the ground state and the first excited S state of the hydrogen atom, to an accuracy of 10^–15E_h. In addition, we have freely optimized the exponents in basis sets containing up to 12 Gaussians. We have studied the convergence of the total energy, the kinetic energy, the extent of the atom as measured by r², and the Fermi-contact interaction at the nucleus in these basis sets as well as in basis sets augmented with additional diffuse or steep functions.     

Electronic coupling calculation and pathway analysis of electron transfer reaction using ab initio fragment-based method. I. FMO-LCMO approach

By making use of an ab initio fragment-based electronic structure method, fragment molecular orbital-linear combination of MOs of the fragments (FMO-LCMO), developed by Tsuneyuki et al. [Chem. Phys. Lett. 476, 104 (2009)], we propose a novel approach to describe long-distance electron transfer (ET) in large system. The FMO-LCMO method produces one-electron Hamiltonian of whole system using the output of the FMO calculation with computational cost much lower than conventional all-electron calculations. Diagonalizing the FMO-LCMO Hamiltonian matrix, the molecular orbitals (MOs) of the whole system can be described by the LCMOs. In our approach, electronic coupling T(DA) of ET is calculated from the energy splitting of the frontier MOs of whole system or perturbation method in terms of the FMO-LCMO Hamiltonian matrix. Moreover, taking into account only the valence MOs of the fragments, we can considerably reduce computational cost to evaluate T(DA). Our approach was tested on four different kinds of model ET systems with non-covalent stacks of methane, non-covalent stacks of benzene, trans-alkanes, and alanine polypeptides as their bridge molecules, respectively. As a result, it reproduced reasonable T(DA) for all cases compared to the reference all-electron calculations. Furthermore, the tunneling pathway at fragment-based resolution was obtained from the tunneling current method with the FMO-LCMO Hamiltonian matrix.     

Calculation of isotropic Compton profiles with Gaussian basis sets

In this paper we present an adaptive algorithm for calculating the isotropic Compton profile (ICP) for any type of Gaussian basis set. The ICP is a measure of the momentum density of electrons and it can be obtained from inelastic X-ray scattering experiments employing synchrotron radiation. We have performed calculations of the ICP for water and helium monomers and dimers using density-functional theory, Hartree-Fock and post-Hartree-Fock methods, with Dunning-type ((d-)aug-)cc-p(C)VXZ basis sets. We have examined the convergence of the Compton profile as a function of the basis set and the level of theory used for the formation of the density matrix. We demonstrate that diffuse basis functions are of utmost importance to the calculation of Compton profiles. Basis sets of at least triple-ζ quality appended by diffuse functions should be used in Compton profile calculations in order to obtain sufficient convergence with regard to the current, experimentally feasible accuracy for systems consisting of light elements.     

Perturbational approach to the electron correlation cusp applied to helium‐like atoms

A recently proposed perturbational approach to the electron correlation cusp problem 1 is tested in the context of three spherically symmetrical two‐electron systems: helium atom, hydride anion, and a solvable model system. The interelectronic interaction is partitioned into long‐ and short‐range components. The long‐range interaction, lacking the singularities responsible for the electron correlation cusp, is included in the reference Hamiltonian. Accelerated convergence of orbital‐based methods for this smooth reference Hamiltonian is shown by a detailed partial wave analysis. Contracted orbital basis sets constructed from atomic natural orbitals are shown to be significantly better for the new Hamiltonian than standard basis sets of the same size. The short‐range component becomes the perturbation. The low‐order perturbation equations are solved variationally using basis sets of correlated Gaussian geminals. Variational energies and low‐order perturbation wave functions for the model system are shown to be in excellent agreement with highly accurate numerical solutions for that system. Approximations of the reference wave functions, described by fewer basis functions, are tested for use in the perturbation equations and shown to provide significant computational advantages with tolerable loss of accuracy. Lower bounds for the radius of convergence of the resulting perturbation expansions are estimated. The proposed method is capable of achieving sub‐μHartree accuracy for all systems considered here. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B-Ne, and Al-Ar

Correlation consistent basis sets have been optimized for use with explicitly correlated F12 methods. The new sets, denoted cc-pVnZ-F12 (n=D,T,Q), are similar in size and construction to the standard aug-cc-pVnZ and aug-cc-pV(n+d)Z basis sets, but the new sets are shown in the present work to yield much improved convergence toward the complete basis set limit in MP2-F12/3C calculations on several small molecules involving elements of both the first and second row. For molecules containing only first row atoms, the smallest cc-pVDZ-F12 basis set consistently recovers nearly 99% of the MP2 valence correlation energy when combined with the MP2-F12/3C method. The convergence with basis set for molecules containing second row atoms is slower, but the new DZ basis set still recovers 97%-99% of the frozen core MP2 correlation energy. The accuracy of the new basis sets for relative energetics is demonstrated in benchmark calculations on a set of 15 chemical reactions.     

A perturbation-based super-CI approach for the orbital optimization of a CASSCF wave function

A perturbation theory-based algorithm for the iterative orbital update in complete active space self-consistent-field (CASSCF) calculations is presented. Following Angeli et al. (J. Chem. Phys. 2002, 117, 10525), the first-order contribution of singly excited configurations to the CASSCF wave function is evaluated using the Dyall Hamiltonian for the determination of a zeroth-order Hamiltonian. These authors employ an iterative diagonalization of the first-order density matrix including the first-order correction arising from single excitations, whereas the present approach uses the single-excitation amplitudes directly for the construction of the exponential of an anti-Hermitian matrix resulting in a unitary matrix which can be used for the orbital update. At convergence, the single-excitation amplitudes vanish as a consequence of the generalized Brillouin's theorem. It is shown that this approach in combination with direct inversion of the iterative subspace (DIIS) leads to very rapid convergence of the CASSCF iteration procedure. © 2019 Wiley Periodicals, Inc.     

Scaling in second-order electron correlation calculations using systematic sequences of even-tempered basis sets

Improved results can often be obtained from second-order Rayleigh-Schrödinger perturbation calculations of electron correlation energies using large basis sets by introducing a scaling factor in the zero-order Hamiltonian. The scaling parameter may be determined from full third-order calculations using a smaller basis set. This scaling procedure can be applied in a systematic fashion by employing a sequence of even-tempered basis sets. Calculations illustrating this approach for the beryllium atom and the neon atom are presented. The scaling procedure is also employed in conjunction with a universal systematic sequence of basis functions. Calculations illustrating this Correlation energy — Mang-body perturbation theory.Work supported in part by S.R.C. Research Grant GR/B/4738.6.S.R.C. Advanced Fellow.     

On the topology of the electron density of ${\mathrm {H}}_{3}^{+}$

The topology of the electron density ρ(r) of \({\mathrm {H}}_{3}^{+}\) is revisited by series of ultra fine tuned geometry optimizations within Hartree-Fock self-consistent virial scaling (SCVS) approach in combination with correlation consistent cc-pVXZ basis sets. The calculations are extended to approach the Hartree-Fock complete basis set (CBS) limit. It is discussed that within such tuned ab initio calculations, the sources of errors that are mapped to the final density matrix in normal calculations are essentially eliminated. The results of electron density analysis on such error-free ρ(r) function via the quantum theory of atoms in molecules (QTAIM) confirm unambiguously the non-nuclear attractor (NNA) as the fundamental topological building block (together with three H atomic basins) to describe the bonding in \({\mathrm {H}}_{3}^{+}\) ion-molecule. The convergence patterns of the values of different density-dependent properties toward CBS limit are also explored. It is reported that the cc-pVXZ sets are not only energy-consistent but also density-consistent. Therefore, on the basis of this important density consistency behavior, the CBS limit values of different atomic and bonding indexes are estimated and ultimately the structure and bonding pattern of \({\mathrm {H}}_{3}^{+}\) are concluded.     

Ab initio calculation of electronic circular dichroism fortrans-cyclooctene using London atomic orbitals

Summary The second-quantization magnetic dipole operator that arises when London atomic orbitals are used as basis functions is derived. In atomic units, the magnetic dipole operator is defined as the negative of the first derivative of the electronic Hamiltonian containing the interaction with the external magnetic field. It is shown that for finite basis sets, the gauge origin dependence of the resulting magnetic dipole operator is analogous to that of the exact operator, and that the derived operator converges to the exact operator in the limit of a complete basis set. It is also demonstrated that the length expression for the rotatory strength in linear response calculations gives gauge-origin-independent results. Sample calculations ontrans-cyclooctene and its fragments are presented. Compared to conventional orbitals, the basis set convergence of the rotatory strengths calculated in the length form using London atomic orbitals is favourable. The rotatory strength calculated fortrans-cyclooctene agrees nicely with the corresponding experimental circular dichroism spectrum, but the spectra for the fragment molecules show little resemblance with that oftrans-cyclooctene.Dedicated to Prof. Jan Linderberg     

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.     

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.     

Large-amplitude dynamics in vinyl radical: the role of quantum tunneling as an isomerization mechanism

We report tunneling splittings associated with the large amplitude 1,2 H-atom migration to the global minima in the vinyl radical. These are obtained using a recent full-dimensional ab initio potential energy surface (PES) [A. R. Sharma, B. J. Braams, S. Carter, B. C. Shepler, and J. M. Bowman, J. Chem. Phys. 130(17), 174301 (2009)] and independently, directly calculated "reaction paths." The PES is a multidimensional fit to coupled cluster single and double and perturbative treatment of triple excitations coupled-cluster single double triple (CCSD(T)) with the augmented correlation consistent triple zeta basis set (aug-cc-pVTZ). The reaction path potentials are obtained from a series of CCSD(T)/aug-cc-pVnTZ calculations extrapolated to the complete basis set limit. Approximate 1D calculations of the tunneling splitting for these 1,2-H atom migrations are obtained using each of these potentials as well as quite different 1D Hamiltonians. The splittings are calculated over a large energy ranges, with results from the two sets of calculations in excellent agreement. Though negligibly slow (>1 s) for the vibrational ground state, this work predicts tunneling-promoted 1,2 hydride shift dynamics in vinyl to exhibit exponential growth with internal vibrational excitation, specifically achieving rates on the sub-μs time scale at energies above E ≈ 7500 cm(-1). Most importantly, these results begin to elucidate the possible role of quantum isomerization through barriers without dissociation, in competition with the more conventional picture of classical roaming permitted over a much narrower window of energies immediately below the bond dissociation limit. Furthermore, when integrated over a Boltzmann distribution of thermal energies, these microcanonical tunneling rates are consistent with sub-μs time scales for 1,2 hydride shift dynamics at T > 1400 K. These results have potential relevance for combustion modeling of low-pressure flames, as well as recent observations of nuclear spin statistical mixing from high-resolution IR/microwave spectroscopy on vinyl radical.     

Basis-set quality and basis-set bias in molecular property calculations

Quality measures for Gaussian basis sets are proposed that are based on principal angles between the basis set and reference molecular orbitals. The principal angles are obtained from the cosine-sine (CS) decomposition of orthogonal matrices and yield detailed information about basis-set convergence with respect to different regions of space. Principal angles for occupied orbitals show excellent correlation with basis-set errors in ground-state energies. Furthermore, ground-state bias in finite basis sets can be estimated from the relation between principal angles for occupied and Rydberg orbitals. Ground-state bias is observed in basis sets including extensive diffuse augmentation and affects the quality of computed molecular response properties. Principal angles and ground-state bias are investigated for the H-Ne atoms and a series of diatomics using numerical Hartree-Fock calculations as a reference. Convergence of ground-state energies and static polarizabilities is studied for the hierarchies of correlation-consistent and Karlsruhe segmented def2 basis sets including different levels of diffuse augmentation.     

Property-optimized gaussian basis sets for molecular response calculations

With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H-Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree-Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree-Fock polarizabilities. The mean absolute basis set errors are 3.6%, 1.1%, and 0.3% for property-optimized basis sets of split-valence, triple-zeta, and quadruple-zeta valence quality, respectively. Density functional and second-order M?ller-Plesset polarizabilities show similar basis set convergence. We demonstrate the efficiency of our basis sets by computing static polarizabilities of icosahedral fullerenes up to C(720) using hybrid density functional theory.     

Estimates of electronic coupling for excess electron transfer in DNA

Electronic coupling V(da) is one of the key parameters that determine the rate of charge transfer through DNA. While there have been several computational studies of V(da) for hole transfer, estimates of electronic couplings for excess electron transfer (ET) in DNA remain unavailable. In the paper, an efficient strategy is established for calculating the ET matrix elements between base pairs in a pi stack. Two approaches are considered. First, we employ the diabatic-state (DS) method in which donor and acceptor are represented with radical anions of the canonical base pairs adenine-thymine (AT) and guanine-cytosine (GC). In this approach, similar values of V(da) are obtained with the standard 6-31G(*) and extended 6-31+ +G(**) basis sets. Second, the electronic couplings are derived from lowest unoccupied molecular orbitals (LUMOs) of neutral systems by using the generalized Mulliken-Hush or fragment charge methods. Because the radical-anion states of AT and GC are well reproduced by LUMOs of the neutral base pairs calculated without diffuse functions, the estimated values of V(da) are in good agreement with the couplings obtained for radical-anion states using the DS method. However, when the calculation of a neutral stack is carried out with diffuse functions, LUMOs of the system exhibit the dipole-bound character and cannot be used for estimating electronic couplings. Our calculations suggest that the ET matrix elements V(da) for models containing intrastrand thymine and cytosine bases are essentially larger than the couplings in complexes with interstrand pyrimidine bases. The matrix elements for excess electron transfer are found to be considerably smaller than the corresponding values for hole transfer and to be very responsive to structural changes in a DNA stack.     

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.