Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

Irreducible tensor analysis of sum- and difference-frequency generation in partially oriented samples

The macroscopic non-liner optical susceptibility _X⁽²⁾ for a system of partially oriented molecules is written as the ensemble average of molecular hyperpolarizability tensors β. The orientational distribution of these molecules is described by a probability function expanded in Wigner matrices. For a rotationally invariant system _X⁽²⁾ has seven non-vanishing components, and a method is outlined to determine these by sum- or difference-frequency mixing experiments. The method could be applied to electrically poled samples, adsorbates liquid crystals and membranes to determine several components of the molecular hyperpolarizability tensor, or to extract information about the orientational distribution.     

Young operator methods for fermion systems

Summary Alternative methods to the standard Young technique for the construction of Fermion wave functions in the spin orbital formalism are presented and shown to be equivalent to the standard technique. To develop these methods: (i) the starting or primitive function is factored into spin and spatial parts, (ii) the conjugacy feature required to satisfy the antisymmetry principle is exploited, (iii) the necessary commutation relations with the Fermion antisymmetrizer are shown to hold and (iv) the one-to-one correspondence between the independent picture of the Young tableaux and the independent Slater determinants is used. This last feature has the advantage of reducing all three methods to rapid efficient graphical procedures. Each method is analyzed to consider the amount of labor involved to carry it out. Several examples of the methods are given for constructing both electronic wave functions and spin functions.     

Vector coupling coefficients for calculations of transition-metal atoms and ions by the SCF coupling operator method

We derived the necessary conditions to which the vector coupling coefficients (VCC ) a and b describing atomic L,S-multiplets of the configurations d^N (1 ≤ N ≤ 9), should satisfy. Special attention is paid to the states of non-Roothaan type for which VCC depend on the choice of degenerate d-orbitals basis set determined within the accuracy up to an orthogonal transformation u. It is shown that for such states the direct sum of matrices ‖a‖ and ‖b‖ must be the non-symmetric matrix. Obtained VCC were used for the ab initio calculations (basis set (14s9p5d)/[8s4p2d] from [15]) on first-row transition atoms (from Sc to Cu) to compare to similar calculations [16], in which the Peterson's VCC have been used, and with calculations [15] carried out by the atomic SCF program [4] as well.     

Complete nuclear motion Hamiltonian in the irreducible normal mode tensor operator formalism for the methane molecule

A rovibrational model based on the normal-mode complete nuclear Hamiltonian is applied to methane using our recent potential energy surface [A. V. Nikitin, M. Rey, and Vl. G. Tyuterev, Chem. Phys. Lett. 501, 179 (2011)]. The kinetic energy operator and the potential energy function are expanded in power series to which a new truncation-reduction technique is applied. The vibration-rotation Hamiltonian is transformed systematically to a full symmetrized form using irreducible tensor operators. Each term of the Hamiltonian expansion can be thus cast in the tensor form whatever the order of the development. This allows to take full advantage of the symmetry properties for doubly and triply degenerate vibrations and vibration-rotation states. We apply this model to variational computations of energy levels for (12)CH(4), (13)CH(4), and (12)CD(4).     

Symplectic splitting operator methods for the time-dependent Schrodinger equation

We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of the system allows us to build highly efficient symplectic integrators at any order. The new methods are accurate, easy to implement, and very stable in comparison with other standard symplectic integrators.     

Comments on calculations of nuclear spin-spin coupling constants using the blinder operator

It is pointed out that the finite second-order hyperfine self-coupling energies, obtained by the Blinder operator, are unphysical and 10² times too large. The additional terms in the first-order matrix element are also unphysical. Therefore the Blinder operator gives no improvement over the simple delta function. The consequences on the recent calculations by Paviot and Hoarau and by Sänger and Voitländer are discussed.     

Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

Unitary group tensor operator algebras for many-electron systems: II. One- and two-body matrix elements

Relying on our earlier results in the unitary group Racah-Wigner algebra, specifically designed to facilitate quantum chemical calculations of molecular electronic structure, the tensor operator formalism required for an efficient evaluation of one- and two-body matrix elements of molecular electronic Hamiltonians within the spin-adapted Gel'fand-Tsetlin basis is developed. Introducing the second quantization-like creation and annihilation vector operators at the unitary group [U(n)] level, appropriate two-box symmetric and antisymmetric irreducible tensor operators as well as adjoint tensors are defined and their matrix elements evaluated in the electronic Gel'fand-Tsetlin basis as single products of segment values. Using these tensor operators, the matrix elements of one- and two-body components of a general electronic Hamiltonian are found. Explicit expressions for all relevant quantities pertaining to at most two-column irreducible representations that are required in molecular electronic structure calculations are given. Relationships with other approaches and possible future extensions of the formalism to partitioned bases or spin-dependent Hamiltonians are discussed.On leave from: Department of Chemistry, Xiamen University, Xiamen, Fujian, PR China.     

Experimental and quantum-chemical determination of the (2)H quadrupole coupling tensor in deuterated benzenes

Deuterium Quadrupole Coupling Constant (DQCC) in benzene was determined both experimentally by Nuclear Magnetic Resonance spectroscopy in Liquid Crystalline solutions (LC NMR) and theoretically by ab initio electronic structure calculations. DQCCs were measured for benzene-d(1) and 1,3,5-benzene-d(3) using several different liquid crystalline solvents and taking vibrational and deformational corrections into account in the analysis of experimental dipolar couplings, used to determine the orientational order parameter of the dissolved benzene. The experimental DQCC results for the isotopomers benzene-d(1) and 1,3,5-benzene-d(3) are found to be 187.7 kHz and 187.3 kHz, respectively, which are essentially equal within the experimental accuracy (+/-0.4 kHz). Theoretical results were obtained at different C-D bond lengths, and by applying corrections for electron correlation and rovibrational motion on top of large-basis-set Hartree-Fock results. The computations give a consistent DQCC of ca. 189 kHz for three different isotopomers; benzene-d(1), 1,3,5-benzene-d(3), and benzene-d(6), revealing that isotope effects are not detectable within the present experimental accuracy. Calculations carried out using a continuum solvation model to account for intermolecular interaction effects result in very small changes as compared to the data obtained in vacuo. The comparison of theoretical and experimental results points out the selection of the underlying molecular geometry as the most likely source of the remaining discrepancy of less than 2 kHz. Such an agreement between the calculated and the experimental DQCC results can only be achieved if rovibrational effects are considered on one hand in the experimental direct dipolar coupling data, and on the other hand in the theoretical property calculation, as is done presently.     

Improving the calculation of magnetic coupling constants in MRPT methods

The magnetic coupling in transition metal compounds with more than one unpaired electron per magnetic center has been studied with multiconfigurational perturbation theory. The usual shortcomings of these methodologies (severe underestimation of the magnetic coupling) have been overcome by describing the Slater determinants with a set of molecular orbitals that maximally resemble the natural orbitals of a high‐level multiconfigurational reference configuration interaction calculation. These orbitals have significant delocalization tails onto the bridging ligands and largely increase the coupling strengths in the perturbative calculation. © 2014 Wiley Periodicals, Inc.     

Excitonic coupling in polythiophenes: comparison of different calculation methods

In conjugated polymers the optical excitation energy transfer is usually described as Forster-type hopping between so-called spectroscopic units. In the simplest approach using the point-dipole approximation the transfer rate is calculated based on the interaction between the transition dipoles of two spectroscopic units. In the present work we compare this approach with three others: The line-dipole approximation, the Coulomb integral between the transition densities, and a quantum-chemical calculation of the interacting dimer as entity. The latter two approaches are based on the semiempirical method ZINDO. The line-dipole approximation is an attractive compromise between computational effort and precision for calculations of the excitonic coupling in extended conjugated polymers.     

Analysis of the convergence of the general coupling operator method for one-configuration-type wave functions

We comment on the convergence of the general coupling operator for all types of one-configuration or multiconfigurational wave functions that still preserve the one-configuration structure for the energy expression. The choice on the best arbitrary real and antisymmetric parameters inherent in the coupling operator methodology is discussed, giving a theoretical reason. Another type of coupling operator is defined, presented, and analyzed. Finally, we give some numerical examples related to the low-lying electronic states of a cluster model for K₂NiF₄ solid. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 368–376, 1998     

Irreducible representation and projection operator application to understanding nonlinear optical phenomena: hyper-Raman, sum frequency generation, and four-wave mixing spectroscopy

Symmetry plays an essential role in understanding optical activities of a molecule in infrared and Raman vibrational spectroscopy as well as in nonlinear optical vibrational spectroscopy. Each vibrational mode belongs to an irreducible representation of the underlying symmetry group. In this paper, using the alpha-helical polypeptide symmetry as an example, we calculate all the third rank nonzero hyper-Raman tensors as well as the infrared and Raman tensors by applying the projection operators to each irreducible species. We demonstrate that the projection operator method provides selection rules for the infrared, Raman, and hyper-Raman vibrational transitions and also other nonlinear optical spectroscopy such as sum frequency generation and the four-, five-, and six-wave mixing coherent vibrational transitions. Specific expressions for all nonzero elements of the corresponding nonlinear susceptibility tensors in a laboratory-fixed coordinate frame are also deduced.     

Nickel-catalyzed alkyl coupling reactions: evaluation of computational methods

The B3LYP, M06, M06L, M062X, MPW1K, and PBE1PBE DFT methods were evaluated for modeling nickel-catalyzed coupling reactions. The reaction consists of a nucleophilic attack by a carbanion equivalent on the nickel complex, S(N)2 attack by the anionic nickel complex on an alkyl halide, and reductive elimination of the coupled alkane product, regenerating the nickel catalyst. On the basis of CCSD(T)//DFT single-point energies, the B3LYP, M06, and PBE1PBE functionals were judged to generate the best ground state geometries. M06 energies are generally comparable or superior to B3LYP and PBE1PBE energies for transition state calculations. The MP2 and CCSD methods were also evaluated for single-point energies at the M06 geometries. The rate-determining step of this reaction was found to be nucleophilic attack of a L(2)NiR anion on the alkyl halide.     

Efficient and accurate approximations to the molecular spin-orbit coupling operator and their use in molecular g-tensor calculations

Approximations to the Breit-Pauli form of the spin-orbit coupling (SOC) operator are examined. The focus is on approximations that lead to an effective quasi-one-electron operator which leads to efficient property evaluations. In particular, the accurate spin-orbit mean-field (SOMF) method developed by Hess, Marian, Wahlgren, and Gropen is examined in detail. It is compared in detail with the "effective potential" spin-orbit operator commonly used in density functional theory (DFT) and which has been criticized for not including the spin-other orbit (SOO) contribution. Both operators contain identical one-electron and Coulomb terms since the SOO contribution to the Coulomb term vanishes exactly in the SOMF treatment. Since the DFT correlation functional only contributes negligibly to the SOC the only difference between the two operators is in the exchange part. In the SOMF approximation, the SOO part is equal to two times the spin-same orbit contribution. The DFT exchange contribution is of the wrong sign and numerically shown to be in error by a factor of 2-2.5 in magnitude. The simplest possible improvement in the DFT-SOC treatment [Veff(-2X)-SOC] is to multiply the exchange contribution to the Veff operator by -2. This is verified numerically in calculations of molecular g-tensors and one-electron SOC constants of atoms and ions. Four different ways of handling the computationally critical Coulomb part of the SOMF and Veff operators are discussed and implemented. The resolution of the identity approximation is virtually exact for the SOC with standard auxiliary basis sets which need to be slightly augmented by steep s functions for heavier elements. An almost as efficient seminumerical approximation is equally accurate. The effective nuclear charge model gives results within approximately 10% (on average) of the SOMF treatment. The one-center approximation to the Coulomb and one-electron SOC terms leads to errors on the order of approximately 5%. Small absolute errors are obtained for the one-center approximation to the exchange term which is consequently the method of choice [SOMF(1X)] for large molecules.     

The determination and assignment of a complete 14N quadrupole coupling tensor in liquid solution

¹⁴N line splittings in the spectrum of nitrobenzene (neat liquid) and metadinitrobenzene (dissolved in benzene), induced by an external electric field, have been used to determine the complete ¹⁴N quadrupole coupling tensor of these substances. Assuming that both molecules are rigid and planar, and that the quadrupole coupling tensors at the ¹⁴N nuclei are identical, the principal components in a local reference frame (x′, y′, z′) are (eQ/h)V_x′ x′ = ±0.34 MHz (eQ/h)V_y′_y′= ±1.18 Hz and (eQ/h)V_z′_z′ = ±1.52. The z′-direction is parallel to the CN bond and the y?direction is perpendicular to the plane of the nitrogroup. With these data the asymmetry parameter η = 0.55.     

The N quadrupole coupling tensor of pyridine and pyrimidine in solution, assigned by NMR in an electric field

The principal components of the ¹⁴N quadrupole coupling tensor in liquid pyridine and pyrimidine are determined from the ²H and ¹⁴N NMR spe