Berry Phase and Spin-Coupled Valence Bond Wavefunction for H3
H3 is a system of three electrons moving in the electrostatic field of three (stationary) protons.
The quantum state of these three electrons is described by a wavefunction phi(r1,r2,r3,s1,s2,s3), where
  • r1, r2, r3 are electron positions in R^3;
  • s1, s2, s3 are electron spins (up or down) in the direction perpendicular to the plane of the protons.
    • Hence the wavefunction is a vector of eight (complex valued, square integrable) functions of nine real variables:
    phi(r1,r2,r3,up,up,up)
    phi(r1,r2,r3,up,up,down)
    phi(r1,r2,r3,up,down,up)
    phi(r1,r2,r3,up,down,down)
    phi(r1,r2,r3,down,up,up)
    phi(r1,r2,r3,down,up,down)
    phi(r1,r2,r3,down,down,up)
    phi(r1,r2,r3,down,down,down).
    Suppose A[ ] is the antisymmetrization operator in the three compound variables x1=(r1,s1), x2=(r2,s2), x3=(r3,s3)

    A[ g(x1,x2,x3) ] = 1/6*[ g(x1,x2,x3) - g(x2,x1,x3) + g(x2,x3,x1) - g(x3,x2,x1) + g(x3,x1,x2) - g(x1,x3,x2) ].

    The Pauli Exclusion Principle also requires that A[ phi ] = phi, i.e. the electrons are indistinguishable Fermions.

    The enormous complexity of such a wavefunction is tamed when it can be adequately approximated
    by a more simple wavefunction which allows an intuitive visualization of the electronic state,
    where individual electrons are located in orbitals, and the spins of these electrons are coupled in defined ways.
    This simplified Spin-Coupled Valence Bond Wavefunction (SCVB, see references below) is of the form

    phi(r1,r2,r3,s1,s2,s3) = A[ psi1(r1) * psi2(r2) * psi3(r3) * Theta(s1,s2,s3) ],

    where
  • psi1(r), psi2(r), psi3(r) are one-electron orbitals, not assumed to be mutually orthogonal;
  • Theta(s1,s2,s3) is the total spin function;

    • The interpretation is that one electron is in the orbital psi1, another electron in psi2, and another in psi3,
    and the spins of the electrons in psi1, psi2, and psi3 are `coupled' according to the total spin function
    Theta(s1,s2,s3). This interpretation has not yet been demonstrated to be of phi, but only of a particular way of generating phi.
    A given phi might have more than one interpretation. However we will speak of the interpretation as if it were a feature of phi.

    A total spin function Theta(s1,s2,s3) can be thought of as a vector with eight complex components.
    The vectors for Theta(s2,s1,s3) and Theta(s1,s3,s2) can be written as the product of 8x8 permutation
    matrices P12 and P23 times the vector for Theta(s1,s2,s3).
    Total spin functions are expanded in terms of tensor products of one-electron spin functions a(s) and b(s)
  • a(up)=1, a(down)=0.
  • b(up)=0, b(down)=1.
    • A tensor product such as a(s1)a(s2)b(s3), abbreviated as aab, is represented by the vector [0,1,0,0,0,0,0,0]^T.
    There are two observable quantities, both represented by 8x8 Hermitian matricies, that are relevant to spins.
    One, denoted by S^2, represents the squared length of the net 3D spin angular momentum vector.
    The second, denoted by S_z, represents the z component of the net 3D spin angular momentum vector.
    (The plane of the nuclei is the xy plane, so the z direction is perpendicular to that plane.)
    These two matrices commute with each other, and hence can be simultaneously diagonalized.
    S^2 has eigenvalues 3/4 and 15/4, whereas S_z has eigenvalues -3/2, -1/2, 1/2, 3/2.
    The joint eigenspaces (which are orthogonal to each other) have the following dimensions:
    (S^2,S_z)=(15/4,-3/2), dimension 1.
    (S^2,S_z)=(15/4,-1/2), dimension 1.
    (S^2,S_z)=(15/4,1/2), dimension 1.
    (S^2,S_z)=(15/4,3/2), dimension 1.
    (S^2,S_z)=(3/4,-1/2), dimension 2.
    (S^2,S_z)=(3/4,1/2), dimension 2.
    S^2 and S_z also commute with each of the two 8x8 permutation matrices P12 and P23.
    Even though the H3 electronic Hamiltonian does not involve any of the spin variables
    the Pauli exclusion principle forces any wavefunction in the first four joint eigenspaces
    to have a significantly larger energy than energy minimizers in the last two joint eigenspaces.
    This is (roughly) because in the first four joint eigenspaces there is an oriented line in R^3 along which
    all three spins are positively aligned (as opposed to a pair being antiparallel);
    in the first and fourth joint eigenspaces this line is parallel to the z axis.
    This forces the electrons away from each other to a higher degree than the balance of electrostatic
    attraction (electrons for protons) and repulsion (electrons for electrons) would have it.
    Energy minimizers in the last two joint eigenspaces have the same energy.
    The total spin function in the 6th joint eigenspace is Theta=c1*Kotani1+c2*Kotani2 where:
  • Kotani1=[2aab-aba-baa]*6^(-1/2)=[(aab-aba)+(aab-baa)]*6^(-1/2). (An equal quantum superposition of
    two states: aab-aba, where electrons in orbitals 2 and 3 are singlet coupled and the electron in orbital 1 is spin up,
    and aab-baa, where electrons in orbitals 1 and 3 are singlet coupled and the electron in orbital 2 is spin up.)
  • Kotani2=[aba-baa]*2^(-1/2). (Electrons in orbitals 1 and 2 are singlet coupled; electron in orbital 3 is spin up.)
    • Optimized values of c1 and c2 will be called spin coupling coefficients.
    Kotani1 and Kotani2 are eigenvectors of P12 with eigenvalues 1 and -1 respectively.
    P23*Kotani1=-1/2*Kotani1+sqrt(3)/2*Kotani2, and P23*Kotani2=sqrt(3)/2*Kotani1+1/2*Kotani2.

    The nuclei are numbered as P1 (top), P2 (lower left), P3 (lower right). (These labels are not shown in the orbital figures below.)
    ...............P1
    ............../..\
    ............./....\
    ............/......\
    .........../........\
    ......l3./..........\.l2
    ........./............\
    ......../..............\
    ......./................\
    .....P2-------------P3
    ..............l1
    The geometry of the nuclei is specified via three numbers, (l1^2,l2^2,l3^2)
    which are squared edge lengths [bohr^2] of the triangle,
    in the following order: P2-P3 (bottom), P3-P1 (right diagonal), P1-P2 (left diagonal).
    A sequence of 13 geometries forming a cycle in shape space around the geometry (4,4,4) are as follows:
    01) (4.06,3.97,3.97), isosceles, l1 long, B2
    02) (4.05,4.00,3.95), scalene
    03) (4.03,4.03,3.94), isosceles, l3 short, A1
    04) (4.00,4.05,3.95), scalene
    05) (3.97,4.06,3.97), isosceles, l2 long, B2
    06) (3.95,4.05,4.00), scalene
    07) (3.94,4.03,4.03), isosceles, l1 short, A1
    08) (3.95,4.00,4.05), scalene
    09) (3.97,3.97,4.06), isosceles, l3 long, B2
    10) (4.00,3.95,4.05), scalene
    11) (4.03,3.94,4.03), isosceles, l2 short, A1
    12) (4.05,3.95,4.00), scalene
    13) (4.06,3.97,3.97), isosceles, l1 long, B2
    The ith geometry is ( 4+0.06*cos(t_i) , 4+0.03*(-cos(t_i)+sqrt(3)*sin(t_i)) , 4-0.03*(cos(t_i)+sqrt(3)*sin(t_i)) ),
    where t_i=(i-1)*pi/12, i=1,2,...,13. The 13th geometry is the same as the first.

    For each geometry the lowest energy wavefunction of SCVB type is shown. The energy in atomic units is shown.
    The figures show psi1(r), psi2(r), psi3(r), where r is restricted to the plane of the nuclei, in horizontal rows.
    Each one-electron orbital is pictured as a 3D graph and as a contour graph side-by-side.
    The symmetry group and the irreducible representation of the three-electron wavefunction is also indicated.
    (See further comments and references after the figures.)
    Geometry01: (4.06,3.97,3.97), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 1.
    Geometry01: (4.06,3.97,3.97), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 2.
    Geometry01: (4.06,3.97,3.97), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 3.
    Geometry02: (4.05,4.00,3.95), Spin function=0.2001*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry02: (4.05,4.00,3.95), Spin function=0.2001*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry02: (4.05,4.00,3.95), Spin function=0.2001*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry03: (4.03,4.03,3.94), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 1.
    Geometry03: (4.03,4.03,3.94), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 2.
    Geometry03: (4.03,4.03,3.94), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 3.
    Geometry04: (4.00,4.05,3.95), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry04: (4.00,4.05,3.95), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry04: (4.00,4.05,3.95), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry05: (3.97,4.06,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 1.
    Geometry05: (3.97,4.06,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 2.
    Geometry05: (3.97,4.06,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 3.
    Geometry06: (3.95,4.05,4.00), Spin function=-0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry06: (3.95,4.05,4.00), Spin function=-0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry06: (3.95,4.05,4.00), Spin function=-0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry07: (3.94,4.03,4.03), Spin function=-0.0084*Kotani1+0.99996*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 1.
    Geometry07: (3.94,4.03,4.03), Spin function=-0.0084*Kotani1+0.99996*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 2.
    Geometry07: (3.94,4.03,4.03), Spin function=-0.0084*Kotani1+0.99996*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 3.
    Geometry08: (3.95,4.00,4.05), Spin function=0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry08: (3.95,4.00,4.05), Spin function=0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry08: (3.95,4.00,4.05), Spin function=0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry09: (3.97,3.97,4.06), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 1.
    Geometry09: (3.97,3.97,4.06), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 2.
    Geometry09: (3.97,3.97,4.06), Spin function=0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 3.
    Geometry10: (4.00,3.95,4.05), Spin function=0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry10: (4.00,3.95,4.05), Spin function=0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry10: (4.00,3.95,4.05), Spin function=0.1997*Kotani1+0.9798*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry11: (4.03,3.94,4.03), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 1.
    Geometry11: (4.03,3.94,4.03), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 2.
    Geometry11: (4.03,3.94,4.03), Spin function=0.0084*Kotani1+0.99997*Kotani2, Energy=-1.537635, Symmetry Group: C2v, Irrep: A1, Orbital 3.
    Geometry12: (4.05,3.95,4.00), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 1.
    Geometry12: (4.05,3.95,4.00), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 2.
    Geometry12: (4.05,3.95,4.00), Spin function=-0.1935*Kotani1+0.9811*Kotani2, Energy=-1.537726, Symmetry Group: Cs, Irrep: A', Orbital 3.
    Geometry13: (4.06,3.97,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 1.
    Geometry13: (4.06,3.97,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 2.
    Geometry13: (4.06,3.97,3.97), Spin function=-0.2539*Kotani1+0.9672*Kotani2, Energy=-1.537786, Symmetry Group: C2v, Irrep: B2, Orbital 3.
    Comments about the Berry Phase (Holonomy) Factor
  • These geometries were analyzed sequentially. For geometries 02 through 13 we used the best wavefunction
    from the previous geometry as the starting guess for the energy minimization for the current geometry.
  • As one proceeds through the 13 geometries the orbitals and the spin coupling coefficients are observed
    to vary continuously. Since geometries 01 and 13 are identical we should compare these two wavefunctions.
  • Let phi_01=A[psi1(r1)*psi2(r2)*psi3(r3)*(0.25*Kotani1+0.97*Kotani2)] denote the wavefunction for geometry 01.
  • Visual inspection shows that phi_13=A[psi2(r1)*psi1(r2)*(-1)*psi3(r3))*(-0.25*Kotani1+0.97*Kotani2)] is the
    wavefunction computed for geometry 13, related to phi_01 by parallel translation around the cycle of geometries.
  • By the following property of the antisymmetrization operator:

    phi_*= A[ psi2(r1)*psi1(r2)*psi3(r3)*{c1*Kotani1(s1,s2,s3)+c2*Kotani2(s1,s2,s3)} ]
    = -A[ psi2(r2)*psi1(r1)*psi3(r3)*{c1*Kotani1(s2,s1,s3)+c2*Kotani2(s2,s1,s3)} ]
    = -A[ psi1(r1)*psi2(r2)*psi3(r3)*{c1*Kotani1(s1,s2,s3)-c2*Kotani2(s1,s2,s3)} ]
    =A[ psi1(r1)*psi2(r2)*psi3(r3)*{-c1*Kotani1(s1,s2,s3)+c2*Kotani2(s1,s2,s3)} ]

    we have that ( notice that in phi_13 the third orbital is (-1)*psi3 )

    phi_13= -A[psi1(r1)*psi2(r2)*psi3(r3)*(0.25*Kotani1+0.97*Kotani2)]= -phi_01.

    This is the Berry phase (holonomy) factor of -1 seen concretely.
    phi_* is an example of a 3-electron wavefunction with two (technically) different interpretations.
    • Comments about Wavefunction Spatial Symmetry
  • All the odd numbered geometries are isosceles triangles, and hence the three-electron wavefunction has
    symmetry group C2v. In all cases the orbitals are symmetric with respect to reflection in the plane of the nuclei.
    The two equal sides of an isosceles triangle determine an angle in the plane of the nuclei. The C2 axis bisects this angle.
    Reflection in the plane through the C2 axis and perpendicular to the plane of the nuclei is the symmetry operation sigma_v.
    The atom at the vertex of this angle (bisected by the C2 axis) will be called the vertex atom.
    The side of the triangle opposite the vertex atom (perpendicular to the C2 axis) will be called the opposite side.
    The wavefunction can be either symmetric (irrep A1) or antisymmetric (irrep B2) with respect to the operation sigma_v.
  • In the geometries 1, 5, 9, and 13, orbitals 1 and 2 are invariant under sigma_v, whereas orbital 3 is transformed
    into its negative. This implies that the overall wavefunction transforms according to the irrep B2.
    Either orbital 1 or orbital 2 is localized on the vertex atom and the other orbital from this pair is symmetrically
    distributed over the other two atoms, across the opposite (longest) side. Orbital 3 is concentrated on the nonvertex
    atoms, with a nodal line along the C2 axis.
  • These geometries give rise to the most complex total spin functions we encounter for any of the geometries, i.e.
    0.2539*Kotani1+0.9672*Kotani2=-0.7876*baa+0.5803*aba+0.2073*aab, or
    -0.2539*Kotani1+0.9672*Kotani2=-0.5803*baa+0.7876*aba-0.2073*aab.
    These spin functions have a 62% chance of the electron in the orbital concentrated on the vertex atom being spin down
    (the other two electrons being spin up and occupying two colocated orthogonal orbitals),
    a 34% chance of the electron in the orbital equally distributed over the other two atoms being spin down
    (the other two being spin up and occupying two partially separated orthogonal orbitals),
    and only a 4% chance of the electron in the orbital with the nodal line being spin down
    (the other two being spin up but occupying two partially separated nonorthogonal orbitals).
  • In the geometries 3, 7, and 11, the operation sigma_v appears to transform orbitals 1,2,3 into orbitals 2,1,3.
    Under a transposition of orbitals 1 and 2 a spin function c1*Kotani1+c2*Kotani2 should be transformed into
    -c1*Kotani1+c2*Kotani2 if the wavefunction is to stay the same. Hence these overall wavefunctions will be invariant
    under sigma_v (irrep A1) if and only if c1=0. However in geometries 3, 7, 11 we have computed c1=+/-0.0084.
  • These calculations were done only imposing Cs symmetry in all geometries. The one-electron basis functions did not
    all transform according to an irreducible representation of the group C2v in the odd numbered geometries.
    Therefore sigma_v will not exactly transpose orbitals 1 and 2, but only approximately do so.
    In geometry 7 the one-electron basis functions should have allowed that orbitals 1 and 2 be exactly transposed
    by the action of sigma_v. However the contour graphs of orbitals 1 and 2 show them not to have this property.
    Thus it appears that the computed spin coupling coefficients c1 and c2 are only accurate to within +/- 0.01 or so,
    probably due to incomplete convergence (stopping criteria based only on the energy and its gradient).
  • Thus in geometries 3, 7, and 11 the spins are coupled as if there were a covalent bond across the opposite
    (shortest) side. Orbitals 1 and 2 (the two orbitals are involved in the `covalent bond') are concentrated primarily on
    one or the other of the non vertex atoms with slight delocalization toward the other atom in this pair;
    these orbitals have a more significant delocalization toward the vertex atom, which is an unusual feature
    indicating a high degree of strain. Orbital 3 is entirely concentrated on the vertex atom.
    This SCVB wavefunction can be understood to be essentially a strained H2+H system,
    where the H2 bond is stretched and the H atom is close to the middle of the H2 `molecule'.
    The equilibrium bond length in H2 is about 1.4 bohr whereas in these geometries the H-H distance is about 2 bohr.
    Readers unfamiliar with SCVB wavefunctions may expect orbitals 1 and 2 to be the same if electrons 1 and 2
    are involved in a `covalent bond', but this is only a feature of the Hartree-Fock (HF) approximation,
    which is the SCVB wavefunction with psi1=psi2, psi1 and psi3 are othogonal, and Theta(s1,s2,s3)=Kotani2(s1,s2,s3).
    The HF approximation is pretty accurate for the Lithium atom, but for these H3 configurations significant
    energy improvements can be made by relaxing these HF constraints to yield the SCVB approximation.
    • Comments about the Degenerate Ground State at the Geometry (4,4,4)
  • At the equilateral triangle geometry (4,4,4) there are two linearly independent wavefunctions with the same
    (ground state) energy. All the geometries displayed here are visually indistinguishable from this equilateral traingle.
    The wavefunctions given at geometries 1 and 7 are very close (visually identical) to a linearly independent pair
    of ground state wavefunctions at the geometry (4,4,4). These wavefunctions are orthogonal because they transform
    according to two different irreducible representations (B2 and A1) of the symmetry group C2v. Neither the orbital
    shapes nor the spin functions change appreciably between the geometries 1 (or 7) and the geometry (4,4,4).
  • Other bases of this degenerate ground state eigenspace can be obtained from the pair of geometries 5, 11,
    and the pair of geometries 9, 3.
  • In each of these pairs (1,7), (5,11), and (9,3) of geometries, the wavefunction for the first excited state
    for one of the geometries is visually identical to the wavefunction for the other member of the pair. The spin
    function for the first excited state for one member of the pair is given by the spin function for the wavefunction
    pictured for the other member of the pair.
    • Acknowledgements
  • The electronic structure calculations were performed using Molpro, version 2009.1, a package of ab initio
    programs, written by H.-J. Werner, P. J. Knowles, F. R. Manby, M. Schutz, et al. See Molpro home .
  • The spin coupled valence bond wavefunctions were computed within Molpro using the CASVB module
    written by D. L. Cooper, T. Thorsteinsson, J. Gerratt, et. al. See CASVB references and manual .
  • The primary author Daniel Dix was profoundly aided in utilizing Molpro by Steven Mielke Ph.D.,
    Department of Chemistry, University of Minnesota.
  • The one-electron orbitals were visualized using Molden, version 4.6, a pre- and post processing program
    of molecular and electronic structure, written by G. Schaftenaar. See Molden homepage .
  • The natural orbitals of the valence bond wavefunction were exported in Molden format using a utility
    preexisting within Molpro for this purpose. This Molden format file was then transformed into one describing
    the valence bond one-electron orbitals using a Fortran program written by D. L. Cooper, kindly provided
    to the primary author by Professor Cooper.
  • The computations were performed at the Minnesota Supercomputing Institute ( MSI home ) funded
    by a grant to the primary author from the Institute for Mathematics and Its Applications ( IMA home ).
    The primary author also acknowledges financial and other research support from the IMA
    during its Year on Mathematical Chemistry, Sept. 2008-May 2009.
  • The primary author also acknowledges useful conversations with Mark Herman Ph.D.,
    Department of Mathematics, University of Rochester.