Volume 115, Issue 20, 22 November 2001
 ARTICLES

 Theoretical Methods and Algorithms

A practical method for the use of curvilinear coordinates in calculations of normalmodeprojected displacements and Duschinsky rotation matrices for large molecules
View Description Hide DescriptionWhile use of curvilinear coordinates such as bond lengths and bond angles is common in accurate spectroscopic and/or scattering calculations for triatomic and other small molecules, their use for large molecules is uncommon and restricted. For large molecules, normalmode analysis is feasible but gives sensible results only if the dynamical or spectroscopic process being considered involves changes in angular coordinates, including ring deformations, which are so small that the motion can be approximated by its tangential component. We describe an approximate method by which curvilinear normalmodeprojected displacements and hence Franck–Condon factors, reorganization energies, and vibronic coupling constants, as well as Duschinsky (Dushinsky, Duschinskii) rotation matrices, can be evaluated for large systems. Three illustrative examples are provided: (i) to understand the nature of the first excited state of water, illustrating properties of largeamplitude bending motions; (ii) to understand the nature of the “boat” relaxation of the first excited state of pyridine, illustrating properties of largeamplitude torsional motions; and (iii) to understand the coupling of vibrational modes to the oxidation of bacteriochlorophylla, a paradigm with many applications to both chemical and biological electron transfer, illustrating properties of macrocyclic deformations. The method is interfaced to a wide variety of computational chemistry computer programs.

The pair density functional of the kinetic energy and its simple scaling property
View Description Hide DescriptionFor electronic systems, a simple property of the recently introduced kinetic energy T as a functional of the pair density derived. Approximate explicit expressions for are presented.

Polarization consistent basis sets: Principles
View Description Hide DescriptionThe basis set convergence of Hartree–Fock energies for the HF, and molecules is analyzed using optimized basis functions. Based on these analysis a sequence of polarization consistent basis sets are proposed which should be suitable for systematically improving Hartree–Fock and density functional energies. Analogous to the correlation consistent basis sets designed for correlation energies, higher angular momentum functions are included based on their energetical importance. In contrast to the correlation consistent basis sets, however, the importance of higher angular momentum functions decreases approximately geometric, rather than arithmetic. It is shown that it is possible to design a systematic sequence of basis sets for which results converge monotonic to the Hartree–Fock limit. The primitive basis sets can be contracted by a general contraction scheme. It is found that polarization consistent basis sets provide a faster convergence than the correlation consistent basis sets. Results obtained with polarization consistent basis sets can be further improved by extrapolation.

Rightangle wiggling electron paramagnetic resonance spectroscopy
View Description Hide DescriptionRightangle wiggling (RAW) electron paramagnetic resonance(EPR)spectroscopy is introduced. In this twodimensional pulse EPR technique the orientation of the external magnetic field is varied during the pulse sequence. For this purpose an additional sinusoidal field of variable amplitude is applied perpendicular to the orientation of RAWEPR is an alternative and experimentally much less demanding technique to rightangle spinning for performing anisotropyresolvedEPRexperiments [Sierra and Schweiger, Mol. Phys. 95, 973 (1998)]. It can be applied to improve the resolution of EPR spectra of orientationally disordered systems or to separate overlapping singlecrystalEPR spectra, and to facilitate an unambiguous interpretation of these spectra. A detailed theoretical description of RAWEPR is given. Model calculations for systems with axial and orthorhombic symmetry, as well as a number of RAWEPR experiments on paramagnetic systems with anisotropic interactions are presented.

Interfacing relativistic and nonrelativistic methods. IV. One and twoelectron scalar approximations
View Description Hide DescriptionTwo approximations for the inclusion of scalar relativistic effects into otherwise nonrelativistic molecular selfconsistent field calculations are described, which continue the sequence of approximations from previous work. In the first, the twoelectron contributions are confined to those where each small component density is onecenter. In the second, the twoelectron contributions are approximated by using the bare Coulomb operator, an approach which parallels the Douglas–Kroll–Hess method. The results show that both approximations yield results close to the full twoelectron scalar relativistic method, within 0.001 Å in bond lengths and 0.1 kcal/mol in dissociation energies. The approximations also give results which are very similar to those of the Douglas–Kroll–Hess method. The second approximation is similar in cost to the Douglas–Kroll–Hess method and only contains simple operators whose integrals are easy to evaluate.

The Poisson equation in density fitting for the KohnSham Coulomb problem
View Description Hide DescriptionA new density fitting approach to the Coulomb problem in Kohn–Sham and Hartree–Fock theory is introduced. Almost all of the 2 and 3index repulsion integrals become simple overlaplike integrals, without approximation. The method is tested on numerous benchmark problems, which reveal that accuracy equal to or better than standard density fitting can be achieved with the evaluation of around a tenth of the number of Coulomb integrals. The scaling properties of the method are illustrated for polyalanine helices up to

A unified scheme for ab initio molecular orbital theory and path integral molecular dynamics
View Description Hide DescriptionWe present a general approach for accurate calculation of chemical substances which treats both nuclei and electrons quantum mechanically, adopting ab initio molecular orbitaltheory for the electronic structure and path integral molecular dynamics for the nuclei. The present approach enables the evaluation of physical quantities dependent on the nuclear configuration as well as the electronic structure, within the framework of Born–Oppenheimer adiabatic approximation. As an application, we give the path integral formulation of electric response properties—dipole moment and polarizability, which characterize the changes both in electronic structure and nuclear configuration at a given temperature when uniform electrostatic field is present. We also demonstrate the calculation of a water molecule using the present approach and the result of temperature and isotope effects is discussed.

Modification of nonrelativistic Gaussian basis sets for relativistic calculations
View Description Hide DescriptionA simple method is proposed in which basis sets of Gaussiantype functions (GTFs), suitable for relativistic Dirac–Fock–Roothaan (DFR) calculations, are derived from their nonrelativistic analogs. The relativistic basis set is obtained through augmenting the nonrelativistic one by several GTFs determined from relativistic calculations for hydrogenlike atoms. The usefulness and reliability of the method is illustrated by DFR calculations of the groundstate energies of lanthanide and actinide atoms.

Selfconsistent Ornstein–Zernike approximation compared with Monte Carlo results for twodimensional lattice gases
View Description Hide DescriptionThe selfconsistent Ornstein–Zernike approach (SCOZA) is solved numerically for a lattice gas or Ising model on the simple square lattice in two dimensions. Interactions of varying range are considered, and the results are compared with corresponding simulation ones. We focus especially upon the location of the critical temperature which is identified with the maximum of the specific heat. The maximum remains finite for the finitesized simulation sample and also for SCOZA, which treats infinite lattices in two dimensions as though they were finite samples. We also investigate the influence of the precise form of the interaction, first using an interaction that extends the nearestneighbor case in a simple way and then considering the squarewell interactions used in the simulations. We find that the shift in away from its meanfield value is governed primarily by the range of interaction. Other specific features of the interaction leave a smaller influence but are relevant to a quantitative comparison with simulations. The SCOZA yields accurate results, and the influence of the precise form of the attractive interaction plays a significant role in SCOZA’s success.

Calculating free energies using average force
View Description Hide DescriptionA new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels [Mol. Phys. 98, 773 (2000)]. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples — rotation around the C–C bond of 1,2dichloroethane immersed in water and transfer of fluoromethane across the waterhexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.

Hyperspherical harmonics for tetraatomic systems
View Description Hide DescriptionA recursion procedure for the analytical generation of hyperspherical harmonics for tetraatomic systems, in terms of roworthonormal hyperspherical coordinates, is presented. Using this approach and an algebraic Mathematica program, these harmonics were obtained for values of the hyperangular momentum quantum number up to 30 (about 43.8 million of them). Their properties are presented and discussed. Since they are regular at the poles of the tetraatomic kinetic energy operator, are complete, and are not highly oscillatory, they constitute an excellent basis set for performing a partial wave expansion of the wave function of the corresponding Schrödinger equation in the strong interaction region of nuclear configuration space. This basis set is, in addition, numerically very efficient and should permit benchmarkquality calculations of statetostate differential and integral cross sections for those systems.

Quantummechanical reaction rate constants from centroid molecular dynamics simulations
View Description Hide DescriptionIt has been shown recently that in order for realtime correlation functions obtained from centroid molecular dynamics (CMD) simulations to be directly related, without further approximations, to the corresponding quantum correlation functions, one of the operators should be linear in the position and/or momentum [Jang and Voth, J. Chem. Phys. 111, 2357 (1999)]. Standard reaction rate theory relates the rate constant to the flux–Heaviside or the flux–flux correlation functions, which involve two nonlinear operators and therefore cannot be calculated via CMD without further approximations. We present an alternative, and completely equivalent, reaction rate theory which is based on the position–flux correlation function. The new formalism opens the door to more rigorously using CMD for the calculation of quantum reaction rate constants in general manybody systems. The new method is tested on a system consisting of a doublewell potential bilinearly coupled to a harmonic bath. The results obtained via CMD are found to be in good agreement with the numerically exact results for a wide range of frictions and temperatures.

Distance and entropy for density matrices
View Description Hide DescriptionDensity matrices for systems with a finite number of states are considered as elements in a vector space. A density matrix that is a convex combination of density matrices that are unitary transformations of some initial density matrix will be closer to, or possibly at the same distance from, the density matrix for the ensemble in which all states are equally occupied, compared with the initial density matrix. This distance is correlated, for fewstate systems, with a function simply related to the entropy. This function increases (entropy decreases) with distance from the equaloccupancy density matrix. Other properties of the function are also established.

Hybrid exchangecorrelation functional determined from thermochemical data and ab initio potentials
View Description Hide DescriptionMultiplicative potentials, appropriate for adding to the nonmultiplicative fractional orbital exchange term in the Kohn–Sham equations, are determined from correlated ab initio electron densities. The potentials are examined graphically and are used in conjunction with conventional thermochemical data to determine a new hybrid exchangecorrelation functional, denoted B972. Calculations using B972 are compared with those from (a) the B971 functional [J. Chem. Phys. 109, 6264 (1998)], which has the same functional form and fraction of orbital exchange, but was fitted to just thermochemical data; and (b) the widely used B3LYP functional [J. Chem. Phys. 98, 5648 (1993)]. B972 atomization energies are close to those from B971; total electronic energies and ionization potentials are less accurate, but remain an improvement over B3LYP. Molecular structures from all three functionals are comparable. Static isotropic polarizabilities improve from B3LYP to B971 to B972; the B972 functional underestimates experimental values, which is consistent with the neglect of zeropoint vibrational corrections. NMR shielding constants—determined as the conventional second derivative of the electronic energy—improve from B3LYP to B971 to B972. Shieldings determined directly from these DFT electron densities using the recently proposed MKS approach [Chem. Phys. Lett. 337, 341 (2001)] are two to three times more accurate than the conventional shieldings, and exhibit an analogous improvement across the three functionals. Classical reaction barriers for sixteen chemical reactions improve significantly from B3LYP to B971 to B972. The introduction of multiplicative potentials into semiempirical hybrid functional development therefore appears beneficial.

Role of molecular distortions in the spin–orbit coupling between the singlet and triplet states of the 4π electron systems and
View Description Hide DescriptionHow molecular distortions enhance the strength of spin–orbit coupling (SOC) between the singlet and triplet states of cyclobutadiene, cyclopentadienyl cation, and cyclopropenyl anion is described. The crossing region of the two potential energy surfaces of cyclobutadiene is characterized by a Jahn–Teller active vibrational mode that can connect the singlet and triplet structures. The spin inversion from triplet to singlet occurs in cyclobutadiene with a structural change from to but the inplane distortion along the Jahn–Teller mode cannot directly enhance the strength of SOC. Molecular distortions along some C–H outofplane bending modes significantly strengthen the SOC in cyclobutadiene. Also in cyclopentadienyl cation, C–H outofplane distortions play an essential role in enhancing the strength of SOC. The outofplane motions destroy the planarity of cyclobutadiene and cyclopentadienyl cation, leading to rehybridization of their σ and π orbitals. This is a main reason that the strength of SOC is enhanced by the C–H outofplane bending distortions in these planar molecules. On the other hand, in cyclopropenyl anion the carbonring distortion that can connect the triplet and singlet structures is a main factor that dominates the transition between the two states, due to its nonplanarity.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Ionization potentials, electron affinities, and vibrational frequencies of neutrals and charged ions from density functional theory
View Description Hide DescriptionGeometrical and electronic properties of neutrals, cations, and anions have been investigated using the density functional method of Becke’s threeparameter hybrid functional with the Perdew/Wang 91 expression. Berny structural optimization and frequency analyses are performed with the basis of for both the neutrals and charged ions. Cohesive energies,ionization potentials, and electron affinities calculated at the optimized groundstate structures agree satisfactorily with recent experimental values. Frequency analyses indicate that the bicapped antitetragonal prism, which was previously proposed as the groundstate structure of is in fact a firstorder stationary point with an imaginary frequency at The optimized groundstate structure of obtained in this work is a distorted, bicapped antitetragonal prism with the symmetry of It is a typical Jahn–Teller distortion. Prominent chargeinduced structural changes are also determined for and

and its neutral precursors: Similarities and differences
View Description Hide DescriptionThe results of a theoretical study of the anion and its neutral precursors are presented. The hydrogenbonded structures were predicted for both the ionic and neutral complexes. The energetically preferred isomer for however is a species with the direct O–I bond. The relation between the potential energy surfaces for ionic and neutral moieties is evaluated based on their electron affinityproperties.Thermodynamic and spectroscopic (IR) properties of complexes are discussed. The interaction energy decomposition is applied to explore the differences between the nature of bonding within the studied complexes.

Four wave mixing spectroscopy for a multilevel system
View Description Hide DescriptionWe present a theory of the fourwavemixing(FWM)spectroscopy for the multilevel system coupled to the bath with arbitrary time scale, in contrast to most previous work that assumes the fast bath dynamic limit (motionally narrowed limit). Exact quantum mechanical expressions for the relevant nonlinear response tensor and the nonlinear polarization vector are obtained for a model Hamiltonian. Effects of molecular rotation and the polarization directions of the input radiation fields are also fully taken into account. The results show how the cross correlation as well as the autocorrelation functions of the fluctuations in the energygaps or transition frequencies between eigenstates contribute to the FWMspectroscopy. The result is applied to threepulse echo spectroscopy. We examine the results for a variety of model spectral density functions, and recover the results of the previously reported theories as limiting cases.

Wet electrons and how to dry them
View Description Hide DescriptionWe present the formation of hydrated electrons by laser vaporization, and investigate in a Fourier transform ion cyclotron resonance (FTICR) mass spectrometer their destruction by the 300 K black body infrared background radiation. While clusters above decay almost exclusively by fragmentation and loss of ligands, the smaller species also detach electrons, with the relative rates of the two processes exhibiting an interesting alternation. Below they almost exclusively detach electrons, and for the detachment rate is apparently so fast that no clusters are observed in the ICRexperiment. From to a pronounced odd–even alternation between electron detachment and fragmention is observed, which is tentatively attributed to entropic rather than energetic effects.

Ab initio study of spectroscopic and radiative characteristics of ionpair states of the molecule
View Description Hide DescriptionElectronic structure and radiative characteristics of lowlying ionpair states of converging to the limits are studied. Ab initio calculations of potential energy curves for the valence and ionpair states and dipole moments for transitions between them are carried out employing the multireference single and doubleexcitation configuration interaction (MRDCI) method, including spin–orbit coupling. It is shown that the lowest two pairs of the ionpair states arise from an avoided crossing between the and parent states, which leads to notably anharmonic shapes of the corresponding potential curves and their mixed nature. This causes significant radial coupling, resulting in the strongly perturbed character of the and states observed experimentally. In contrast, their gerade counterparts run parallel to one another and exhibit much less perturbation. Spectroscopicproperties of the computed adiabatic curves are in very good agreement with the available experimental data. Dipole moments have been calculated for parallel ionpair–valence state transitions and radiative lifetimes have been obtained for the adiabatic ionpair states. A reanalysis of the experimental bound–free emission spectra from the state [N. K. Bibinov et al., Chem. Phys. 254, 89 (2000)] is given.