SPINS: A collection of algorithms for symbolic generation and transformation of many-electron spin eigenfunctions

Summary SPINS represents a collection of algorithms intended to provide an efficient, robust and easy-to-use quantum-chemical toolbox capable of performing a wide range of operations on spin eigenfunctions in the Rumer, Kotani and Serber spin bases. It includes routines for symbolic generation of the Rumer spin eigenfunctions as linear combinations of elementary spin products, for computing all transformation matrices relating the Rumer, Kotani and Serber spin bases and for calculation of the matrices of the irreducible representations of the symmetric group carried by the Rumer, Kotani and Serber spin eigenfunctions, as well as facilities for interpreting general spin-coupling patterns such as those used in spin-coupled theory. The resulting codes, written in Fortran-77 and available on the Internet (from P.B.Karadakov@Bristol.AC.UK or DLC@Liverpool.AC.UK) are so compact and efficient that they even run on IBM PC-compatible personal computers.     

Generation of genealogical spin eigenfunctions

A method is given for generating the Yamanouchi-Kotani genealogical spin eigenfunctions which requires neither storage of eigenfunctions for smaller numbers of electrons, nor summations of large order, nor explicit use of results from the theory of representations of the symmetric group. An explicit formula is given for the coefficients of expansion in terms of spin products.     

A simple algorithm for the construction of genealogical spin functions

A simple algorithm is given for the construction of spin eigenfunctions according to the genealogical scheme. The method can deal directly with the N-electron problem without any knowledge of the (N-1)-electron spin eigenfunctions. It uses the representation matrices corresponding to the transpositions (k,k + 1), the latter can be written down from the knowledge of the Young tableaux.     

Orthogonal Waller–Hartree spin eigenfunctions

It was shown recently by the present author that the double symmetrization and the double antisymmetrization are essential in the spin-dependent and the spin-free formalisms, respectively, to perform the exclusion of all the unnecessary spin eigenfunctions and the selection of a unique set of linearly independent spin eigenfunctions. The double antisymmetrized Wigner matric basis and the Wigner double symmetrized matric basis are presented in this article for N up to 6. The double symmetrization or the double antisymmetrization also results in a direct expansion method for the calculation of the orthogonal spin coefficients; this direct method does not require the knowledge of the wave functions of the N ? 1 electron system. The modified method of Graebenstetter can also be used to calculate these orthogonal spin coefficients.     

Generalized one-electron spin functions and self-similarity measures

A simple method for construction of eigenfunctions of one-electron spin angular momentum operators from products of the primitive one-electron spin functions is presented. Properties of these functions and their applications to the evaluation of some integrals met in theory of quantum similarity are briefly discussed.     

Direct evaluation of spin representation matrices and ordering of permutation-group elements

A direct and general method is presented for constructing the orthogonal spin representation matrices (irreps) of the permutation group corresponding to the Yammanouchi-Kotani coupling scheme. For arbitrary permutations the irreps are constructed directly from the Young tableaus by a process which is, in general, only quadratic in the number of spin eigenfunctions, but which in actual computations becomes linear on vector computers for moderate sizes of the matrices. We also introduce a graphical representation of the group elements and a universal lexical ordering of permutations. The methods have been implemented and computational examples are presented.     

On the resolution of a double determinant

The resolution of a double determinant into a sum of spin-free orthogonal spin eigenfunctions is presented; its equivalent resolution of a spin function product θ^M₀ is also given. Furthermore, a more general resolution of a Slater determinant into all spin eigenfunctions with S ≥ M is also obtained. Present work has also provided a method to calculate the linear coefficients in the expression of the primitive θ^M_i in terms of the linear combination of spin eigenfunctions X? (S, M).     

Generation of generalized branching diagram spin functions by Schmidt orthogonalization of spin-paired functions

The generalized branching diagram (GBD ) spin representation is defined as the method of sequentially coupling together a number of subsystem spin eigenfunctions using the general rules of angular momentum coupling. It is shown that any GBD representation may also be obtained by Schmidt orthogonalizing a set of cannonical spin–paired (SP ) functions, provided the SP basis is suitably ordered. The ordering procedure used is well suited to computer implementation. This is a generalization of results known in the literature for the Yamanouchi–Kotani and for the Serber spin representations.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Proof of the linear independence of properly selected projected spin eigenfunctions

An inductive proof is given of Löwdin's theorem about the linear independence and completeness of a properly selected set of projected spin eigenfunctions.     

Hamiltonian for the spin quantum Hall effect solution and derivation based on an SU (2) gauge field

A Hamiltonian to describe a spin quantum Hall effect with two types of spin‐orbit coupling is introduced and the eigenfunctions and eigenvalues are obtained for it. It is shown that this Hamiltonian also results by gauging a kinetic energy Hamiltonian by an SU (2) gauge field. The Berry phase is obtained from the model wavefunction for the model and used to define a filling factor. This allows for the calculation of the spin Hall conductivity. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Bound particle with magnetic moment in a space with topological defect

We consider a bounded quantum mechanical particle with spin ?1/2 and a gyromagnetic ratio g, which is placed in a uniform magnetic field, in a space with a linear topological defect. We obtain the exact expressions for eigenfunctions and eigenvalues, using the approach of the continuum theory of defects, and show the dependence on the topological parameters and potential harmonic. Besides, we study the limits case and obtained the results described in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Construction of orbital angular momentum eigenfunctions for many-particle systems by harmonic projection

Formulas are derived which allow the direct construction of total orbital angular momentum eigenfunctions for many-particle systems without the use of Clebsch–Gordan coefficients. One of the equations is closely analogous to Dirac' identity for the total spin operator. This equation describes the action of L² on a function of the particle coordinates in terms of a class operator of the symmetric group and a "contraction operator." A general projection operator for constructing symmetric eigenfunctions of L² is presented.     

Efficient generation of configuration interaction matrix elements

A method for the efficient generation of CI matrix elements over spin eigenfunctions is presented. Practical application of the approach is limited to configurations with about 10 open shells, but the algorithm results in the generation of more than 2200 matrix elements/second on an IBM 370/158 computer including all overhead for a large matrix which contains 50% non-zero elements.     

Spin-coupled calculations based on projected spin eigenfunctions

It is outlined how the utilization of a basis of projected spin eigenfunctions can lead to increased computational efficiency in the evaluation of matrix elements and density matrices in spin-coupled valence bond calculations. Received: 17 September 1997 / Accepted: 23 October 1997     

Spin-adapted wave functions of the serber type and time reversal

A procedure is proposed for generating Serber-type spin eigenfunctions withM _s = 0. The procedure uses the time-reversal invariance of these functions to increase the efficiency and to reduce the storage requirements. Simplifications in calculating the matrix elements of an observable operator which follow from the use of the time-reversal symmetry are briefly discussed.     

Characters of two-row representations of the symmetric group

Characters of irreducible representations (irreps) of the symmetric group corresponding to the two-row Young diagrams, i.e., describing transformation properties of N-electron eigenfunctions of the total spin operators, have been expressed as explicit functions of the number of electrons N and of the total spin quantum number S. The formulas are useful in various areas of theory of many-electron systems, particularly in designing algorithms for evaluation of spectral density moments. © 1997 John Wiley & Sons, Inc.     

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.     

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

The local spin formalism ( 3 ) for computing expectation values 〈S_A · S_B〉 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 {J_AB} 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 metal d orbitals 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 〈S_A · S_B〉 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 priori predictions 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     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.