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

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

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

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

where

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).

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

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 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:

two states:

and

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 figures.)

...............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. |
||||||

to vary continuously. Since geometries 01 and 13 are identical we should compare these two wavefunctions.

wavefunction computed for geometry 13, related to phi_01 by parallel translation around the cycle of geometries.

A[ psi2(r1)*psi1(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.

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

The side of the triangle opposite the vertex atom (perpendicular to the C2 axis) will be called the

The wavefunction can be either symmetric (irrep A1) or antisymmetric (irrep B2) with respect to the operation sigma_v.

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.

0.2539*Kotani1+0.9672*Kotani2=-0.7876*

-0.2539*Kotani1+0.9672*Kotani2=-0.5803*

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).

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.

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).

(shortest) side. This is reflected in the shape of the orbitals. Orbitals 1 and 2 are concentrated primarily on

one or the other of the non vertex atoms with slight delocalization toward the other atom in this pair.

These two orbitals are involved in the `covalent bond'. Orbital 3 is entirely concentrated on the vertex atom.

(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).

and the pair of geometries 9, 3.

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.

programs, written by H.-J. Werner, P. J. Knowles, F. R. Manby, M. Schutz, et al. See Molpro home .

written by D. L. Cooper, T. Thorsteinsson, J. Gerratt, et. al. See CASVB references and manual .

Department of Chemistry, University of Minnesota.

of molecular and electronic structure, written by G. Schaftenaar. See Molden homepage .

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.

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.

Department of Mathematics, University of Rochester.