The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

Stable and efficient algorithms for Xα multiple scattering calculations

Solutions to some practical problems that arise in multiple scattering calculations on large molecules are discussed. (1) Numerical instabilities near the zero of energy can be removed by rescaling the secular matrix. (2) The calculation of structure factors can be made much more efficient by the application of symmetry projection operators. (3) An energy search procedure is described that ensures that no states are inadvertently neglected. Test calculations incorporating these changes illustrate the improved numerical stability, and show decreases in computation time of 30–60%, when compared to previous codes. The procedures suggested here are applicable to both relativistic and nonrelatitistic calculations.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

A new method to generate spin-orbit coupled potential energy surfaces: effective relativistic coupling by asymptotic representation

A new method has been developed to generate fully coupled potential energy surfaces including derivative and spin-orbit coupling. The method is based on an asymptotic (atomic) representation of the molecular fine structure states and a corresponding diabatization. The effective relativistic coupling is described by a constant spin-orbit coupling matrix and the geometry dependence of the coupling is accounted for by the diabatization. This approach is very efficient, particularly for certain systems containing a very heavy atom, and yields consistent results throughout nuclear configuration space. A first application to a diatomic system is presented as proof of principle and is compared to accurate ab initio calculations. However, the method is widely applicable to general polyatomic systems in full dimensionality, containing several relativistic atoms and treating higher order relativistic couplings as well.     

Spin-orbit vibronic coupling in 3Pi states of linear triatomic molecules

The Renner-Teller vibronic-coupling problem of a 3Pi electronic state of a linear molecule is analyzed with the inclusion of the spin-orbit coupling of the 3Pi electronic state, employing the microscopic (Breit-Pauli) spin-orbit coupling operator for the two unpaired electrons. The 6x6 Hamiltonian matrix in a diabatic spin-electronic basis is obtained by an expansion of the molecular Hamiltonian in powers of the bending amplitude. The symmetry properties of the Hamiltonian with respect to the time-reversal operator and the relativistic vibronic angular momentum operator are analyzed. It is shown that there exists a linear vibronic-coupling term of spin-orbit origin, which has not been considered so far in the Renner-Teller theory of 3Pi electronic states. While two of the six adiabatic electronic wave functions do not exhibit a geometric phase, the other four carry nontrivial topological phases which depend on the radius of the integration contour. The spectroscopic effects of the linear spin-orbit vibronic-coupling mechanism have been analyzed by numerical calculations of the vibronic spectrum for selected model examples.     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

Computational strategies for a four-component Dirac-Kohn-Sham program: implementation and first applications

An implementation of the generalized gradient approximation within the four-component formulation of relativistic density-functional theory using G-spinor basis sets is presented. This approach is based on the direct evaluation of the relativistic density and its gradient from the G-spinor amplitudes and gradients without explicit reference to the total density matrix. This proves to be a particularly efficient scheme, with an intrinsic computational cost that scales linearly with the number of G-spinor basis functions. In order to validate this new implementation, incorporated in the parallel version of the program BERTHA, a detailed study of the diatomic system CsAu is also reported. The spectroscopic constants D(e),r(e),omega(e), and x(e)omega(e) and the dipole moment mu have been calculated and compared with the best available theoretical and experimental data. The sensitivity of our results to the details of the numerical schemes used to evaluate the matrix elements is analyzed in detail. Also presented is a comparative study of molecular properties in the alkali auride series which have been obtained using several standard non-relativistic density functionals.     

Recent Development of Relativistic Molecular Theory

Today it is common knowledge that relativistic effects are important in the heavy-element chemistry. The continuing development of the relativistic molecular theory is opening up rows of the periodic table that are impossible to treat with the non-relativistic approach. The most straightforward way to treat relativistic effects on heavy-element systems is to use the four-component Dirac-Hartree-Fock approach and its electron-correlation methods based on the Dirac-Coulomb(-Breit) Hamiltonian. The Dirac-Hartree-Fock (DHF) or Dirac-Kohn-Sham (DKS) equation with the four-component spinors composed of the large- and small-components demands severe computational efforts to solve, and its applications to molecules including heavy elements have been limited to small- to medium-size systems. Recently, we have developed a very efficient algorithm for the four-component DHF and DKS approaches. As an alternative approach, several quasi-relativistic approximations have also been proposed instead of explicitly solving the four-component relativistic equation. We have developed the relativistic elimination of small components (RESC) and higher-order Douglas-Kroll (DK) Hamiltonians within the framework of the two-component quasi-relativistic approach. The developing four-component relativistic and approximate quasi-relativistic methods have been implemented into a program suite named REL4D.In this article, we will introduce the efficient relativistic molecular theories to treat heavy-atomic molecular systems accurately via the four-component relativistic and the two-component quasi-relativistic approaches. We will also show several chemical applications including heavy-element systems with our relativistic molecular approaches.     

Recent Development of Relativistic Molecular Theory

Summary. Today it is common knowledge that relativistic effects are important in the heavy-element chemistry. The continuing development of the relativistic molecular theory is opening up rows of the periodic table that are impossible to treat with the non-relativistic approach. The most straightforward way to treat relativistic effects on heavy-element systems is to use the four-component Dirac-Hartree-Fock approach and its electron-correlation methods based on the Dirac-Coulomb(-Breit) Hamiltonian. The Dirac-Hartree-Fock (DHF) or Dirac-Kohn-Sham (DKS) equation with the four-component spinors composed of the large- and small-components demands severe computational efforts to solve, and its applications to molecules including heavy elements have been limited to small- to medium-size systems. Recently, we have developed a very efficient algorithm for the four-component DHF and DKS approaches. As an alternative approach, several quasi-relativistic approximations have also been proposed instead of explicitly solving the four-component relativistic equation. We have developed the relativistic elimination of small components (RESC) and higher-order Douglas-Kroll (DK) Hamiltonians within the framework of the two-component quasi-relativistic approach. The developing four-component relativistic and approximate quasi-relativistic methods have been implemented into a program suite named REL4D.In this article, we will introduce the efficient relativistic molecular theories to treat heavy-atomic molecular systems accurately via the four-component relativistic and the two-component quasi-relativistic approaches. We will also show several chemical applications including heavy-element systems with our relativistic molecular approaches.     

Relativistic electronic structure theory

The theoretical and technical foundations are presented for the efficient relativistic electronic structure theories to treat heavy-atomic molecular systems. This review contains two surveys of four-component and two-component quasi-relativistic approaches. First, we review our highly efficient computational scheme for four-component relativistic ab initio molecular orbital (MO) methods over generally contracted spherical harmonic Gaussian-type spinors (GTSs). Illustrative calculations, which are performed with a new four-component relativistic ab initio molecular orbital program package REL4D, clearly show the efficiency of our computational scheme by the Dirac-Hartree-Fock (DHF) and Dirac-Hartree-Fock (DKS) methods. Next, in the two-component quasi-relativistic framework, two relativistic Hamiltonians, RESC and higher order Douglas-Kroll (DK) Hamiltonians, are introduced, and several illustrative calculations are shown. Numerical results for several systems show that good accuracy can be obtained with our third-order DK (DK3) Hamiltonian.     

New concepts in bonded functions theory. III. Orthogonalization problem

One of possible approaches to the CI method is based on Boys bonded functions which can be generated in a systematic way forming an independent set of high internal symmetry. The main disadvantage of bonded functions is their nonorthogonality. In this paper a scheme is proposed for passing to orthogonalized set of bonded functions together with the appropriate algorithm for the transformation of the energy matrix H . The orthogonalization matrices are shown to reflect high symmetry of the canonical set of bonded functions, and in what follows they can be defined by short vectors. Moreover, the orthogonalization transformation can be handled in a blockwise manner.     

Integral algorithm and density matrix integration scheme for ab initio band structure calculations on polymeric systems

A new program for band structure calculations of periodic one-dimensional systems has been constructed. It is distinguishable from other codes by the efficient two-electron integral evaluation and the integration schemes of the density matrix in the first Brillouin zone. The computation of polymeric two-electron integrals is based on the McMurchie Davidson algorithm and builds batches of the different cell indices included in the polymeric system. Consequently it presents efficient scaling with respect to the number of unit cells taken into account. Our algorithm takes into account fully the polymeric symmetry rather than the molecular symmetry. A semidirect procedure where only exchange integrals are computed at each SCF cycle is proposed in order to maintain balance between computation time and disk space. In addition, the integration of the density matrix over a large number of cell indices can be performed by different methods, such as Gauss-Legendre, Clenshaw-Curtis, Filon, and Alaylioglu-Evans-Hyslop. This last scheme is able to obtain an accuracy of 10(-13) a.u. on each individual density matrix element for all cell indices with only 48 k-points.     

Symmetric group approach to relativistic CI. I. General formalism

General formulas for matrix elements of spin-dependent operators in a basis of spin-adapted antisymmetrized products of orthonormal orbitals are derived. The resulting formalism may be applied to construction of the Hamiltonian matrices both for Pauli and for projected no-pair relativistic configuration interaction methods. From a formal point of view, it is a generalization of the symmetric group approach to the CI method for the case of spin-dependent Hamiltonians. © 1997 John Wiley & Sons, Inc.     

Theoretical study of Pu and Am tetracarbide molecules

The electronic structure and ground‐state molecular properties of Pu and Am tetracarbides have been investigated by relativistic multireference calculations using CASSCF/CASPT2 theory as well as by density functional theory in conjunction with relativistic pseudopotentials. The CASSCF/CASPT2 treatment has been extended by spin–orbit coupling effects for selected species using the CAS state‐interaction method. The five atoms can form various structural isomers, from which 12 ones have been identified in our study. The electronic ground state in both molecules corresponds to a planar fan‐type structure of C_2v symmetry, in which the actinide atom is connected to a bent C₄ moiety. The other structures are much higher in energy, the ones computed in this study appear between 250 and 1050 kJ/mol. The bonding characteristics in the most relevant structures have been analyzed on the basis of the valence molecular orbitals and natural bond orbital analysis. The most stable structures have been characterized by their spectroscopic (vibrational and electron) properties. © 2014 Wiley Periodicals, Inc.     

Fuzzy Symmetries for Linear Molecules and their Molecular Orbitals

In this paper, the fuzzy symmetry of some prototypical linear molecules has been analyzed. The results show that some molecular orbitals (MOs) are less symmetrical but some others are more symmetrical than the molecular skeleton, which the MOs correspond to. The membership functions of space inversion for MOs are closely related to the chemical characteristics of the MOs. Sometimes, although the symmetry of a molecular skeleton is not obvious, however that of some MO is quite obvious. The membership functions of the fuzzy inversion symmetry depend on the choice of the position of the center of inversion. As compared to those of diatomic molecules and linear tri-atomic molecules, the linear polyatomic molecules in which a distinctive fuzzy symmetry of space translation may exist, and thus a significant effect on their properties can be expected.     

Time-dependent quasirelativistic density-functional theory based on the zeroth-order regular approximation

A time-dependent quasirelativistic density-functional theory for excitation energies of systems containing heavy elements is developed, which is based on the zeroth-order regular approximation (ZORA) for the relativistic Hamiltonian and a noncollinear form for the adiabatic exchange-correlation kernel. To avoid the gauge dependence of the ZORA Hamiltonian a model atomic potential, instead of the full molecular potential, is used to construct the ZORA kinetic operator in ground-state calculations. As such, the ZORA kinetic operator no longer responds to changes in the density in response calculations. In addition, it is shown that, for closed-shell ground states, time-reversal symmetry can be employed to simplify the eigenvalue equation into an approximate form that is similar to that of time-dependent nonrelativistic density-functional theory. This is achieved by invoking an independent-particle approximation for the induced density matrix. The resulting theory is applied to investigate the global potential-energy curves of low-lying LambdaS- and omega omega-coupled electronic states of the AuH molecule. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are in good agreement with those of time-dependent four-component relativistic density-functional theory and ab initio multireference second-order perturbation theory. Nonetheless, this two-component relativistic version of time-dependent density-functional theory is only moderately advantageous over the four-component one as far as computational efforts are concerned.     

A numerical method to obtain a symmetry-adapted basis from the hamiltonian or a similar matrix

A practical method is proposed which using the hamiltonian matrix, or some other matrix corresponding to any operator with identical symmetry properties, enables one to obtain the transformation matrix, from the given basis to a symmetry-adapted basis. The method is very suitable for applications in the fields of molecular orbital and force constant calculations.     

Efficient algorithm for the spin-free valence bond theory. I. New strategy and primary expressions

A new algorithm for nonorthogonal valence bond (VB) method is presented by using symmetric group approach. In the present algorithm, a new function, called paired-permanent-determinant (PPD), is defined, which is an algebrant and has the same symmetry of a corresponding VB structure. The evaluation of a PPD is carried out by using a recursion formula similar to the Laplace expansion method for determinants. An overlap matrix element in the spin-free VB method may be obtained by evaluating a corresponding PPD, while the Hamiltonian matrix element is expressed in terms of the products of electronic integrals and sub-PPDs. In the present work, some important properties of PPDs are discussed, and the primary procedure for the evaluation of PPD is deduced. Furthermore, the expressions for evaluating both the overlap and Hamiltonian matrix elements are also given in details, which are essential to develop an efficient algorithm for nonorthogonal VB calculations. In the present study, some further effective technical considerations will be adopted, and a new ab initio VB program will be introduced. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 287–297, 1998     

A relativistic C.I. study on theJ=1/2 even spectrum of Sr II

We present a method for performing relativistic CI calculations in ground and excited atomic and ionic levels. An electron occupying a relativistic shellnκ in a given electronic configuration is described by a single numerical four-component Dirac-Fock orbital having the samen and κ quantum numbers to those of the shellnκ. Application of this method yields estimates for the I.P. (88 741 cm^?1) and the core correlation energy (?30916 cm^?1) for Sr II and for the total correlation energy in Sr III (?30916 cm^?1). Core-valence correlation energies for the |core 5s〉 (?4379 cm^?1), |core 6s〉 (?1191 cm^?1) and |core 13s〉 (?32 cm^?1) levels have been calculated for Sr II. Estimates for the total relativistic, Breit, vacuum polarization and self energy corrections for these levels are also reported. Configurations in which the core is not fully occupied have been found to yield significant contributions to the correlation energies of both ground and excited levels. Our results show that full scale relativistic CI calculations using numerical four-component Dirac-Fock orbitals are feasible and provide a useful ab-initio tool for the investigation of atomic properties within the framework of fully relativistic theories.     

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

Symmetry simplifications are introduced in configuration interaction (CI ) by reducing the number of symmetry-allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group S_n, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.