Charge transfer interaction in the effective fragment potential method

An approximate formula is derived and implemented in the general effective fragment potential (EFP2) method to model the intermolecular charge transfer interaction. This formula is based on second order intermolecular perturbation theory and utilizes canonical molecular orbitals and Fock matrices obtained with preparative self-consistent field calculations. It predicts charge transfer energies that are in reasonable agreement with the reduced variational space energy decomposition analysis. The formulas for the charge transfer gradients with respect to EFP translational and rotational displacements are also derived and implemented.     

Ab initio calculations of optical absorption spectra: solution of the Bethe-Salpeter equation within density matrix perturbation theory

We describe an ab initio approach to compute the optical absorption spectra of molecules and solids, which is suitable for the study of large systems and gives access to spectra within a wide energy range. In this approach, the quantum Liouville equation is solved iteratively within first order perturbation theory, with a Hamiltonian containing a static self-energy operator. This procedure is equivalent to solving the statically screened Bethe-Salpeter equation. Explicit calculations of single particle excited states and inversion of dielectric matrices are avoided using techniques based on density functional perturbation theory. In this way, full absorption spectra may be obtained with a computational workload comparable to ground state Hartree-Fock calculations. We present results for small molecules, for the spectra of a 1 nm Si cluster in a wide energy range (20 eV), and for a dipeptide exhibiting charge transfer excitations.     

Calculating intensities using effective Hamiltonians in terms of Coriolis-adapted normal modes

The calculation of rovibrational transition energies and intensities is often hampered by the fact that vibrational states are strongly coupled by Coriolis terms. Because it invalidates the use of perturbation theory for the purpose of decoupling these states, the coupling makes it difficult to analyze spectra and to extract information from them. One either ignores the problem and hopes that the effect of the coupling is minimal or one is forced to diagonalize effective rovibrational matrices (rather than diagonalizing effective rotational matrices). In this paper we apply a procedure, based on a quantum mechanical canonical transformation for deriving decoupled effective rotational Hamiltonians. In previous papers we have used this technique to compute energy levels. In this paper we show that it can also be applied to determine intensities. The ideas are applied to the ethylene molecule.     

Application of second-order SCF perturbation theory to the calculation of mixed-frequency hyperpolarizabilities from time-dependent Hartree-Fock theory

Second-order SCF perturbation theory is used to solve the TDHF equations of Dalgarno and Victor through the introduction of frequency dependent density matrices. Exploratory calculations are reported for the frequency dependent polarizability and hyper-polarizability of LiH.     

Comparison of perturbative and variational treatments of molecular vibrations: application to the vibrational spectrum of HFCO up to 8000 cm(-1)

We calculated highly excited states of the HFCO molecule, comparing results from two methods. In the first method, Van Vleck perturbation theory is used to transform away all off-diagonal couplings except those between nearly degenerate states. This perturbative transformation leads to a matrix representation where eigenvalues are obtained with relatively small matrices. In the second method, variational eigenvalues are obtained by combining the Jacobi-Wilson approach with the block-Davidson scheme. The key ingredient here is a prediagonalized-perturbative scheme applied to a subspace of a curvilinear normal-mode basis set. Comparisons of the two methods provide a critical test of the less time-consuming perturbation theory. Two different coordinate sets are used to test the sensitivity of the results to coordinate choice. Perturbation theory also requires a polynomial fit to the potential. The implications of this restriction are investigated.     

Constrained equilibrium as a tool for characterization of deformable porous media

A new method for characterizing the deformable porous materials with noncritical adsorption probes is proposed. The mechanism is based on driving the adsorbate through a sequence of constrained equilibrium states with the insertion isotherms forming a pseudocritical point or a van der Waals-type loop. In the framework of a perturbation theory and Monte Carlo simulations we have found a link between the loop parameters and the host morphology. This allows one to characterize porous matrices through analyzing a shift of the pseudocritical point and a shape of the pseudospinodals.     

Combination of perturbative and variational methods for calculating molecular spectra: calculation of the upsilon=3-5 CH stretch overtone spectrum of CHF3

Highly excited states of the CHF3 molecule belonging to the third, fourth, and fifth Fermi polyad are calculated using a combination of the Van Vleck perturbation theory and a variational treatment. The perturbation theory preconditions the Hamiltonian matrix by transforming away all couplings except those between nearly degenerate states. This transformation is implemented so that eigenvalues can be found with significantly smaller matrices than that which would be needed in the original normal mode representation. Even with preconditioning, at the energies as high as 3-5 quanta in the CH stretch, it is not possible to directly diagonalize the Hamiltonian matrix due to the large basis sets required. Iterative methods, particularly the block-Davidson method, are explored for finding the eigenvalues. The methods are compared and the advantages discussed.     

Frequency dependent nonlinear optical properties of molecules: Formulation and implementation in the HONDO program

This article summarizes the detailed equations for the time-dependent Hartree–Fock treatment of nonlinear properties for perturbations made up of a static electric field and an oscillating field. Explicit expressions for all nonlinear processes up to third order are obtained in terms of the density matrices at the same order. For processes at second and third order in perturbation, expressions in terms of lower order quantities are also obtained by applying the (2n + 1) theorem of perturbation theory. The corresponding computer implementation in the HONDO program is described.     

Unitary perturbation theory: a generalization of two-by-two rotations

A generalization of the two-by-two rotation technique is proposed, permitting a whole row (column) of the matrix to be treated simultaneously. The method is based on the explicit analytical evaluation of the matrix exponent representing a symmetric combination of the individual rotations. Besides constructing the unitary transformation matrices, a new orthogonalization algorithm is also proposed. The resulting “unitary perturbation theory” and orthogonalization method may be useful in different areas. Received: 15 November 1999 / Accepted: 30 January 2000 / Published online: 19 April 2000     

Second-order exchange-induction energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new formulation of the second-order exchange-induction energy of symmetry-adapted perturbation theory is presented. In the proposed formalism the exchange-induction energy is expressed through one- and two-particle reduced density matrices of monomers, which are of zeroth and first order with respect to the effective electrostatic potential of another monomer. The resulting expression is further modified by using the partition of two-particle density matrices into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. The proposed formalism has been applied to the case of closed-shell monomers and for density matrices obtained from the expectation-value expression with coupled cluster singles and doubles wave functions. The performance of the new approach has been demonstrated on several benchmark van der Waals systems, including dimers of argon, water, and ethyne.     

CAST: A new program package for the accurate characterization of large and flexible molecular systems

The presented program package, Conformational Analysis and Search Tool (CAST) allows the accurate treatment of large and flexible (macro) molecular systems. For the determination of thermally accessible minima CAST offers the newly developed TabuSearch algorithm, but algorithms such as Monte Carlo (MC), MC with minimization, and molecular dynamics are implemented as well. For the determination of reaction paths, CAST provides the PathOpt, the Nudge Elastic band, and the umbrella sampling approach. Access to free energies is possible through the free energy perturbation approach. Along with a number of standard force fields, a newly developed symmetry‐adapted perturbation theory‐based force field is included. Semiempirical computations are possible through DFTB+ and MOPAC interfaces. For calculations based on density functional theory, a Message Passing Interface (MPI) interface to the Graphics Processing Unit (GPU)‐accelerated TeraChem program is available. The program is available on request. © 2014 Wiley Periodicals, Inc.     

The vibrational auto-adjusting perturbation theory

In this work a new method to calculate anharmonic vibrational ground and excited state energies is proposed. The method relies on the auto-adjusting perturbation theory (APT) which has been successfully used to diagonalize square matrices. We use as zeroth order correction the self-consistent vibrational energies, and with the APT approach we calculate the vibrational anharmonic correlation correction to any desired order. In this paper we present the methodology and apply it to a model system and formaldehyde. Vibrational APT approach shows a robust convergent behavior even for the states where the standard (Rayleigh-Schrödinger) vibrational Møller-Plesset perturbation theory is clearly divergent.     

Effect of scaling of a quantum mechanical force field on the frequencies and forms of molecular vibrations

The effect of scaling of an ab initio quantum mechanical force field on the frequencies and forms of normal vibrations are studied in terms of first- and second-order perturbation theory. Scaling the force constant matrix according to Pulay using certain assumptions in first-order perturbation theory is equivalent to scaling vibration frequencies and does not modify the form of vibrations. In this case, the second-order corrections to the frequencies and forms of vibrations become zero. The first-order perturbation theory formulas are used to verify the assumptions by calculating the frequencies and matrices of transition to perturbed forms of vibrations of ethane, propane, ethylene, cyclopropene, and isobutene molecules from quantum mechanical force fields found with the 6-31G basis set. It is shown that the vibration frequencies calculated by the formulas of first-order perturbation theory are in good agreement with exact values; the matrix of transition to perturbed eigenvectors is rarefied, with only ≈1% of its elements being markedly nonzero. Moscow State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 210–216, March–April, 1998. This work was supported by RFFR grant No. 96-03-34085.     

A general,recursive, and open‐ended response code

We present a new implementation of a recent open‐ended response theory formulation for time‐ and perturbation‐dependent basis sets (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) at the Hartree–Fock and density functional levels of theory. A novel feature of the new implementation is the use of recursive programming techniques, making it possible to write highly compact code for the analytic calculation of any response property at any valid choice of rule for the order of perturbation at which to include perturbed density matrices. The formalism is expressed in terms of the density matrix in the atomic orbital basis, allowing the recursive scheme presented here to be used in linear‐scaling formulations of response theory as well as with two‐ and four‐component relativistic wave functions. To demonstrate the new code, we present calculations of the third geometrical derivatives of the frequency‐dependent second hyperpolarizability for HSOH at the Hartree–Fock level of theory, a seventh‐order energy derivative involving basis sets that are both time and perturbation dependent. © 2014 Wiley Periodicals, Inc.     

Rayleigh–Schrödinger perturbation theory: Practical implementation of matrix and vector formalisms and description of an heuristic sufficiency convergence criterion

Starting from previous work, where Rayleigh–Schrödinger perturbation theory has been reformulated in matrix form, a practical algorithm implementation is described using both full matrix and vector alternatives. An heuristic convergence sufficiency criterion based on Gershgorin discs is also presented. Some numerical examples related to atomic CI computations are reported to illustrate the theoretical framework.     

Turbo charging time-dependent density-functional theory with Lanczos chains

We introduce a new implementation of time-dependent density-functional theory which allows the entire spectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a single standard ground-state calculation. This method is particularly well suited for large systems and/or large basis sets, such as plane waves or real-space grids. By using a superoperator formulation of linearized time-dependent density-functional theory, we first represent the dynamical polarizability of an interacting-electron system as an off-diagonal matrix element of the resolvent of the Liouvillian superoperator. One-electron operators and density matrices are treated using a representation borrowed from time-independent density-functional perturbation theory, which permits us to avoid the calculation of unoccupied Kohn-Sham orbitals. The resolvent of the Liouvillian is evaluated through a newly developed algorithm based on the nonsymmetric Lanczos method. Each step of the Lanczos recursion essentially requires twice as many operations as a single step of the iterative diagonalization of the unperturbed Kohn-Sham Hamiltonian. Suitable extrapolation of the Lanczos coefficients allows for a dramatic reduction of the number of Lanczos steps necessary to obtain well converged spectra, bringing such number down to hundreds (or a few thousands, at worst) in typical plane-wave pseudopotential applications. The resulting numerical workload is only a few times larger than that needed by a ground-state Kohn-Sham calculation for a same system. Our method is demonstrated with the calculation of the spectra of benzene, C(60) fullerene, and of chlorophyll a.     

Vibrational quasi-degenerate perturbation theory: applications to fermi resonance in CO2, H2CO, and C6H6

A quasi-degenerate perturbation method with vibrational self-consistent field (VSCF) reference wavefunction is developed. It simultaneously accounts for strong anharmonic mode-mode coupling among a few states (static correlation) by a configuration interaction theory and for weak coupling with a vast number of the other states (dynamic correlation) by a perturbation theory. A general formula is derived based on the van Vleck perturbation theory. An algorithm that selects a compact set of the most important VSCF configurations which contribute to the static correlation is proposed and a scheme to limit the number of configurations considered for dynamic correlation is also implemented. This method reproduces the vibrational frequencies of CO2 and H2CO that are subject to the strongest anharmonic mode-mode coupling within 10 cm(-1) of vibrational configuration interaction results in a computational expense reduced by a factor of one to two orders of magnitude. The method also reproduces the infrared absorption of C6H6 in the CH stretching (nu12) frequency region, in which combination tones nu13nu16 and nu2nu13nu18 appear on account of an intensity borrowing from nu12via the anharmonic coupling.     

Orbital-optimized third-order Møller-Plesset perturbation theory and its spin-component and spin-opposite scaled variants: application to symmetry breaking problems

In this research, orbital-optimized third-order M?ller-Plesset perturbation theory (OMP3) and its spin-component and spin-opposite scaled variants (SCS-OMP3 and SOS-OMP3) are introduced. Using a Lagrangian-based approach, an efficient, quadratically convergent algorithm for variational optimization of the molecular orbitals (MOs) for third-order M?ller-Plesset perturbation theory (MP3) is presented. Explicit equations for response density matrices, the MO gradient, and Hessian are reported in spin-orbital form. The OMP3, SCS-OMP3, and SOS-OMP3 approaches are compared with the second-order M?ller-Plesset perturbation theory (MP2), MP3, coupled-cluster doubles (CCD), optimized-doubles (OD), and coupled-cluster singles and doubles (CCSD) methods. All these methods are applied to the O(4)(+), O(3), and seven diatomic molecules. Results demonstrate that the OMP3 and its variants provide significantly better vibrational frequencies than MP3, CCSD, and OD for the molecules where the symmetry-breaking problems are observed. For O(4)(+), the OMP3 prediction, 1343 cm(-1), for ω(6) (b(3u)) mode, where symmetry-breaking appears, is even better than presumably more reliable methods such as Brueckner doubles (BD), 1194 cm(-1), and OD, 1193 cm(-1), methods (the experimental value is 1320 cm(-1)). For O(3), the predictions of SCS-OMP3 (1143 cm(-1)) and SOS-OMP3 (1165 cm(-1)) are remarkably better than the more robust OD method (1282 cm(-1)); the experimental value is 1089 cm(-1). For the seven diatomics, again the SCS-OMP3 and SOS-OMP3 methods provide the lowest average errors, ∣Δω(e)∣ = 44 and ∣Δω(e)∣ = 35 cm(-1), respectively, while for OD, ∣Δω(e)∣ = 161 cm(-1)and CCSD ∣Δω(e)∣ = 106 cm(-1). Hence, the OMP3 and especially its spin-scaled variants perform much better than the MP3, CCSD, and more robust OD approaches for considered test cases. Therefore, considering both the computational cost and the reliability, SCS-OMP3 and SOS-OMP3 appear to be the best methods for the symmetry-breaking cases, based on present application results. The OMP3 method offers certain advantages: it provides reliable vibrational frequencies in case of symmetry-breaking problems, especially with spin-scaling tricks, its analytic gradients are easier to compute since there is no need to solve the coupled-perturbed equations for the orbital response, and the computation of one-electron properties are easier because there is no response contribution to the particle density matrices. The OMP3 has further advantages over standard MP3, making it promising for excited state properties via linear response theory.     

Different forms of perturbation theory for the calculation of the correlation energy

Different forms of perturbation theory for the calculation of correlation energy in both closed-and open-shell systems are discussed. For closed-shell systems, Epstein–Nesbet perturbation theory is compared with Møller–Plesset (MP ) perturbation theory based on canonical Hartree–Fock orbitals and with MP theory based on internally consistent SCF orbitals. The traditional MP theory gives superior results despite its use of an inferior zeroth-order Hamiltonian. This behavior is rationalized in terms of the larger denominators present in the traditional MP theory. These conclusions are used to support the restricted open-shell perturbation methods proposed recently by Murray and Davidson, and these new methods are compared with spin-restricted Epstein–Nesbet theory and the unrestricted MP (UMP ) approach. © 1992 John Wiley & Sons, Inc.     

PERTURB: A special-purpose algebraic manipulation program for classical perturbation theory

Classical perturbation theory provides a particularly promising route to EBK quantization of nonseparable systems. However, the number of terms generated when implementing perturbation theory for systems with more than two degrees of freedom can prove too large for general purpose symbolic manipulators to handle. We describe PERTURB, a specialized algebraic manipulation program written in C for quantization of multidimensional systems. A review of operator based classical perturbation theory is given, and the relationship between this type of perturbation theory and quantum mechanical Van Vleck perturbation theory discussed. The relative performance of the Dragt–Finn and Lie transform algorithms is assessed.