Spin Eigenfunctions Research Articles

A unitary group formulation of open-shell electron propagator theory

By replacing the usual annihilation and creation operators of second quantization by appropriately normalized fundamental Wigner operators of the unitary group U(n) and by representing the many‐electron spin eigenfunctions in terms of the Gelfand–Tsetlin basis for the appropriate irreducible representation, we have succeeded in developing an attractive electron propagator formalism which incorporates closed‐shell, open‐shell, or multiconfigurational reference states. Matrix element evaluation for the fundamental U(n) Wigner operators is treated, and illustrative three‐orbital examples involving reference states of doublet and triplet spin symmetry are presented.

Matrix elements of spin-dependent one-electron operators between bonded functions

Expressions are presented for the matrix elements of a general one-electron spin-dependent operator between bonded functions, i.e., eigenfunctions of total spin constructed with the help of valence-bond or Rumer diagrams. These results should be useful, for example, for the introduction of the spin-orbit operator in configuration interaction calculations of molecules.

Magnetic anisotropy from density functional calculations. Comparison of different approaches: Mn12O12 acetate as a test case

Magnetic anisotropy is the capability of a system in a triplet or higher spin state to store magnetic information. Although the source of the magnetic anisotropy is the zero-field splitting of the ground state of the system, there is a difference between these two quantities that has to be fully rationalized before one makes comparisons. This is especially important for small spins such as triplets, where the magnetic anisotropy energy is only half of the zero-field splitting. Density functional calculations of magnetic anisotropy energies correspond to a high-field limit where the spins are aligned by the external magnetic field. Data are presented for the well-studied molecular magnet Mn(12)O(12) acetate. Both perturbative and self-consistent treatments, different quasirelativistic Hamiltonians (zeroth order regular approximation, Douglas-Kroll, effective core potentials) and exchange-correlation functionals are compared. It is shown that some effects usually considered minor, such as the inclusion of the exchange-correlation potential in the effective one-particle spin-orbit operator, lead to sizable differences when computing magnetic anisotropy energies. Higher-order contributions, that is, the difference between self-consistent and perturbative results, increase the magnetic anisotropy energy somewhat but do not introduce sizeable quartic terms or an in-plane anisotropy. In numerical experiments, on can switch off and on spin-orbit coupling at individual atomic sites. This procedure yields single-site contributions to the overall magnetic anisotropy energy that could be used as parameters in phenomenological spin Hamiltonians. If ferrimagnetic systems are treated with broken symmetry density functional methods where the Kohn-Sham reference function is not a spin eigenfunction, corrections are needed which depend on the size of the exchange couplings in the system and must therefore be evaluated case by case.

Comparison of Semiempirical ZILSH and DFT Calculations of Exchange Constants in Fe4 Butterfly Complexes.

Magnetic interactions in a series of tetranuclear Fe(3+) complexes with the butterfly core structure have been studied with semiempirical ZILSH and density functional theory (DFT) calculations (B3LYP functional). A theoretical analysis of a previously used method of estimating exchange constants from a restricted number of spin configurations reveals systematic errors arising from asymmetry in the complexes, which cause large variations in results with different choices of spin configurations. Correction factors are derived that yield the correct results obtained from full configuration space (FCS) calculations. Exchange constants obtained from DFT FCS calculations for the "body-body" interaction were large and ferromagnetic, in disagreement with values obtained from empirical fits of magnetic susceptibility data for the complexes, established magnetostructural correlations in polynuclear Fe(3+) complexes, and ZILSH calculations. DFT calculations also gave unreasonably large antiferromagnetic exchange constants for interaction between "wingtip" ions that are not directly bridged, again in disagreement with ZILSH calculations. Estimates of exchange constants for interaction of body and wingtip ions obtained with ZILSH and DFT were similar, with the ZILSH values in slightly better agreement with empirical fits. Considering all interactions, the ZILSH method provides results in better accord with experiment than DFT for these complexes. Additional comparisons of exchange constants obtained with different spin coupling schemes showed that values appropriate for two-center spin eigenfunctions gave consistently better results than values calculated with the local spin operator. The effect of basis set was found to be very small. A brief analysis of these findings is given.

Nonlinear wave function expansions: A progress report

AbstractSome recent progress is reported for a novel nonlinear expansion form for electronic wave functions. This expansion form is based on spin eigenfunctions using the Graphical Unitary Group Approach and the wave function is expanded in a basis of product functions, allowing application to closed and open shell systems and to ground and excited electronic states. Each product basis function is itself a multiconfigurational expansion that depends on a relatively small number of nonlinear parameters called arc factors. Efficient recursive procedures for the computation of reduced one‐ and two‐particle density matrices, overlap matrix elements, and Hamiltonian matrix elements result in a very efficient computational procedure that is applicable to very large configuration state function (CSF) expansions. A new energy‐based optimization approach is presented based on product function splitting and variational recombination. Convergence of both valence correlation energy and dynamical correlation energy with respect to the product function basis dimension is examined. A wave function analysis approach suitable for very large CSF expansions is presented based on Shavitt graph node density and arc density. Some new closed‐form expressions for various Shavitt Graph and Auxiliary Pair Graph statistics are presented. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Spin–orbit interaction with nonlinear wave functions

AbstractThe computation of the spin–orbit interaction is discussed for electronic wave functions expressed in the new nonlinear expansion form. This form is based on spin eigenfunctions using the graphical unitary group approach (GUGA). The nodes of a Shavitt graph in GUGA are connected by arcs, and a Configuration State Function (CSF) is represented as a walk along arcs from the vacuum node to a head node. The wave function is a linear combination of product functions each of which is a linear combination of all CSFs, wherein each CSF coefficient is a product of nonlinear arc factors. When the spin–orbit interaction is included the Shavitt graph is a union of single‐headed Shavitt graphs each with the same total number of electrons and orbitals. Thus spin–orbit Shavitt graphs are multiheaded. For full‐CI multiheaded Shavitt graphs, analytic expressions are presented for the number of walks, the number of nodes, the number of arcs, and the number of node pairs in the associated auxiliary pair graph. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Hamiltonian Matrix and Reduced Density Matrix Construction with Nonlinear Wave Functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.

Optimization of nonlinear wave function parameters

AbstractAn energy‐based optimization method is presented for our recently developed nonlinear wave function expansion form for electronic wave functions. This expansion form is based on spin eigenfunctions, using the graphical unitary group approach (GUGA). The wave function is expanded in a basis of product functions, allowing application to closed‐shell and open‐shell systems and to ground and excited electronic states. Each product basis function is itself a multiconfigurational function that depends on a relatively small number of nonlinear parameters called arc factors. The energy‐based optimization is formulated in terms of analytic arc factor gradients and orbital‐level Hamiltonian matrices that correspond to a specific kind of uncontraction of each of the product basis functions. These orbital‐level Hamiltonian matrices give an intuitive representation of the energy in terms of disjoint subsets of the arc factors, they provide for an efficient computation of gradients of the energy with respect to the arc factors, and they allow optimal arc factors to be determined in closed form for subspaces of the full variation problem. Timings for energy and arc factor gradient computations involving expansion spaces of >1024 configuration state functions are reported. Preliminary convergence studies and molecular dissociation curves are presented for some small molecules. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

NMR paramagnetic relaxation due to the S=5∕2 complex, Fe(III)-( tetra-p -sulfonatophenyl)porphyrin: Central role of the tetragonal fourth-order zero-field splitting interaction

The metalloporphyrins, Me-TSPP [Me=Cr(III), Mn(III), Mn(II), Fe(III), and TSPP=meso-(tetra-p-sulfonatophenyl)porphyrin], which possess electron spins S=3/2, 2, 5/2, and 5/2, respectively, comprise an important series of model systems for mechanistic studies of NMR paramagnetic relaxation enhancement (NMR-PRE). For these S>1/2 spin systems, the NMR-PRE depends critically on the detailed form of the zero-field splitting (zfs) tensor. We report the results of experimental and theoretical studies of the NMR relaxation mechanism associated with Fe(III)-TSPP, a spin 5/2 complex for which the overall zfs is relatively large (D approximately = 10 cm(-1)). A comparison of experimental data with spin dynamics simulations shows that the primary determinant of the shape of the magnetic relaxation dispersion profile of the water proton R1 is the tetragonal fourth-order component of the zfs tensor. The relaxation mechanism, which has not previously been described, is a consequence of zfs-induced mixing of the spin eigenfunctions of adjacent Kramers doublets. We have also investigated the magnetic-field dependence of electron-spin relaxation for S=5/2 in the presence of a large zfs, such as occurs in Fe(III)-TSPP. Calculations show that field dependence of this kind is suppressed in the vicinity of the zfs limit, in agreement with observation.

Spin eigenfunctions and operators for the Dn groups

An automated procedure for determining symmetry-adapted spin eigenfunctions is developed for the D n symmetry groups with n equivalent spins for arbitrary size of I and for n=3,…,6. These eigenfunctions are also eigenfunctions of the vector sum of the n equivalent nuclear spins. Generalized hyperfine spin operators are developed that have real matrix elements and that exploit the full symmetry. These spin operators can be combined with operators for other spins and molecular rotation using only real arithmetic.

Ab initio theory of magnetic interactions at surfaces

The low to high spin energy transition of Ni adsorbed on regular and defective sites ofMgO(100) and the relative strengths of bulk and surface magnetic coupling constants offirst row transition metal oxides (MnO, FeO, CoO, NiO and CuO) are taken as examples toillustrate some deficiencies of density functional theory (DFT). For these ionic systems acluster/periodic comparison within the same computational method (either DFT orHartree–Fock) is used to establish that embedded cluster models provide an adequaterepresentation. The cluster model approach is then used to obtain accurate values forthe magnetic properties of interest by using explicitly correlated wavefunctionmethods which handle the electronic open shell rigorously as spin eigenfunctions.

Geminal operators and symmetric requirements

This paper continues our search for the unification of the spin-free and the spin-dependent theories and their required conditions. First, besides the previously uncovered powerful ability of the spin double symmetrizer in the spin part of the primitive Hartree product, we have extended our examination to the additional role played by the spatial geminal (valence bond) operator in the spatial part of the Hartree product . This has thus completed our studies to reveal the important hidden role of the Hartree product. Secondly, we are able to show the capability of the spatial geminal symmetric requirement (GSR) which is the spin-free counterpart of the well-known spin-dependent spin geminal antisymmetric requirement (GAR), and to prove that these two symmetric requirements are interchangeable and either one of them is sufficient. The conditions for these requirements to be satisfied are given. We have also proved that the operation of spin geminal operator on the primitive spin function product is not necessary as it has no effect under the orthogonal spin projection. Its arising difference and similarity with the two symmetric requirements is shown. The spin GAR satisfactory problem of the Yamanouchi Kotani spin eigenfunctions is also explained. An important message is obtained: not all the spatial properties can be replaced by their equivalent supplementary dual spin functions, and vice versa, although these spatial properties may be passed over through the use of the spatial–spin interchangeability theorem (SSIT). An immediate usage of this SSIT to the primitive Hartree product is shown. Finally, various forms of the spin-free and spin-dependent orthogonal spin projectors are produced including those with λ orbital pairs or λ doubly occupied orbitals; they are also interchangeable and exhibit the one-to-one correspondence between the spin-free and the spin-dependent formalisms through our equivalent transformation rule just as expected. Useful relations between these spin projectors and the antisymmetrizers which are helpful for the evaluation of Hamiltonian matrix elements are given.

Explicit expressions for linearly independent spin eigenfunctions

Both the spin-free and the spin-dependent spin eigenfunctions with λ-double occupancy and their reverse relations are produced explicitly. The orthogonal spin coefficients for λ-double occupancy are just some linear sums of 2λ orthogonal spin coefficients for zero-double occupancy. Their linear independence is shown.

Explicit expressions for linearly independent spin eigenfunctions

Both the spin-free and the spin-dependent spin eigenfunctions with λ-double occupancy and their reverse relations are produced explicitly. The orthogonal spin coefficients for λ-double occupancy are just some linear sums of 2λ orthogonal spin coefficients for zero-double occupancy. Their linear independence is shown.

A spin-complete version of the spin-flip approach to bond breaking: What is the impact of obtaining spin eigenfunctions?

Spin-complete versions of the spin-flip configuration-interaction-singles (SF-CIS) approach have been investigated to determine the impact of making the wave function an eigenfunction of Ŝ2. The method has been implemented within an extended restricted active space configuration interaction formalism. Spin-complete results are presented for excitation energies, equilibrium geometries, and potential energy curves for dissociation of a single bond in several small molecules. The effect of different orbital choices has also been investigated. The spin-complete results are compared both to results using the original spin-flip method and to more computationally expensive benchmarks. Using spin eigenfunctions dramatically improves upon the accuracy of the SF-CIS approach.

Semiempirical local spin: Theory and implementation of the ZILSH method for predicting Heisenberg exchange constants of polynuclear transition metal complexes

AbstractThe local spin formalism (Clark, A. E.; Davidson, E. R. J Chem Phys 2001, 115, 7382–7392) for computing expectation values 〈SA· SB〉 that appear in the Heisenberg spin model has been extended to semiempirical single determinant wave functions. An alternative derivation of expectation values in restricted and unrestricted cases is given that takes advantage of the zero differential overlap (ZDO) approximation. A formal connection between single determinant wave functions (which are not in general spin eigenfunctions) and the Heisenberg spin model was established by demonstrating that energies of single determinants that are eigenfunctions of the local spin operators with eigenvalues corresponding to high‐spin radical centers are given by the same Heisenberg coupling constants {JAB} that describe the true spin states of the system. Unrestricted single determinant wave functions for transition metal complexes are good approximations of local spin eigenfunctions when the metaldorbitals are local in character and all unpaired electrons on each metal have the same spin (although spins on different metals might be reversed). Good approximations of the coupling constants can then be extracted from local spin expectation values 〈SA· SB〉 energies of the single determinant wave functions. Once the coupling constants are obtained, diagonalization of the Heisenberg spin Hamiltonian provides predictions of the energies and compositions of the spin states. A computational method is presented for obtaining coupling constants and spin‐state energies in this way for polynuclear transition metal complexes using the intermediate neglect of differential overlap Hamiltonian parameterized for optical spectroscopy (INDO/S) in the ZINDO program. This method is referred to as ZILSH, derived from ZINDO, Davidson's local spin formalism, and the Heisenberg spin model. Coupling constants and spin ground states obtained for 10 iron complexes containing from 2 to 6 metals are found to agree well with experimental results in most cases. In the case of the complex [Fe6O3(OAc)9(OEt)2(bpy)2]+,a prioripredictions of the coupling constants yield a ground‐state spin of zero, in agreement with variable‐temperature magnetization data, and corroborate spin alignments proposed earlier on the basis of structural considerations. This demonstrates the potential of the ZILSH method to aid in understanding magnetic interactions in polynuclear transition metal complexes. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Ab initio spin-orbit CI calculations of the potential curves and radiative lifetimes of low-lying states of lead monofluoride

The electronic structure of the lead monofluoride molecule is studied by means of ab initio configuration interaction (CI) calculations including the spin-orbit interaction. Potential-energy curves are generated for a large number of electronic states, of which only the X1 2Π1/2 ground and X2 2Π3/2 and A 2Σ+ excited states have been observed experimentally. Two different methods are compared for the inclusion of spin-orbit effects in the theoretical treatment, a contracted CI which employs a basis of large-scale Λ–S eigenfunctions to form a rather small matrix representation of the full relativistic Hamiltonian (two-step approach), and a more computationally laborious technique which involves solution of a secular equation of order 250 000 S2 eigenfunctions of different spin and spatial symmetry to achieve a potentially more evenly balanced description of both relativistic and electron correlation effects (one-step approach). In the present application, it is found that both methods achieve quite good agreement with measured spectroscopic constants for the X1, X2, and A states. The simpler of these methods is also employed to predict the radiative lifetimes of the latter two states. The key A 2Σ+–X 2Π transition moment in these calculations is found to vary strongly with internuclear distance and to vanish in the neighborhood of the respective equilibrium distances of both participating states. The computed lifetime for the A, v′=0 state of 16 μs overestimates the corresponding measured value by a factor of three, but those of higher vibrational states are found to decrease rather sharply with increasing v′, suggesting that only a slight displacement of the theoretical A–X transition moment curve is needed to explain the above discrepancy.

Application and development of multiconfigurational localized perturbation theory

Generalization of localized perturbation theory, which results with a method able to span the spin space correctly, is presented. This generalization is achieved by using a multiconfigurational (MC) wave function as the reference. This is the most comprehensive expansion used within MC–LMP2 approach to date, with, however, low computational cost [computational scaling with system size (N) of the new method is O(N3)]. Recently, we have reported the successful Jaguar2 (J2) model for calculating atomization energies. Within the MC–LMP2 framework, the J2 model for calculating heats of formation is based on the generalized valence bond–perfect pairing (GVB–PP) wave function. The J2 model was applied only to closed shell cases because of the perfect pairing (PP) restriction in the reference function. In order to describe other systems, the PP restriction needs to be lifted. This work describes efforts in that direction. The PP restriction can be lifted by a restricted configuration interaction (RCI) procedure applied to the GVB–PP wave function. In this paper, the equations describing the application of LMP2 theory to self-consistent RCI wave function are derived and explained. The RCI wave function is a “true” MC expansion as opposed to the GVB–PP, which uses only a single spin eigenfunction (SEF). We also present the self-consistent (SC) optimization of the RCI wave function. The SC–RCI–LMP2 is the first MC–LMP2 method where the spin space is spanned in the reference. This is important for describing the nondynamical correlation (near degeneracy) effects associated, for example, with bond breaking processes. The SC–RCI–LMP2 is an efficient method applicable to large systems; it is shown to reproduce the potential energy surfaces calculated by the complete active space–second order perturbation (CAS–SCF–PT2) method. This is demonstrated, for the first time, on some widely used test cases.

Many‐electron theory: Density functional approach generalized to treat spin eigenfunctions and relation to spinless low‐order density matrices

Many‐electron theory: Density functional approach generalized to treat spin eigenfunctions and relation to spinless low‐order density matrices

Many-electron theory: Density functional approach generalized to treat spin eigenfunctions and relation to spinless low-order density matrices

Many-electron theory: Density functional approach generalized to treat spin eigenfunctions and relation to spinless low-order density matrices