Title: Algebraic Symmetries in Quantum Chemistry
1Algebraic Symmetries in Quantum Chemistry
Clifford Algebra and Para-Fermi Algebra in
Correlated Many-Electron Theories
- Nicholas D. K. Petraco
- John Jay College and the Graduate Center
- City University of New York
2Outline
- Part I, The Setting
- Quantum Chemistry and many-electron wave
functions - Solving the Schrödinger equation including
electron correlation - Part II, Mathematical Tools
- Representation Theory
- Part III, Application
- Spin-adaptation and some algebra
- Representation theory of the unitary group U(n)
- Para-Fermi algebra
- The Clifford algebra unitary group U(2n)
- U(n) module in U(2n) form
- Matrix element evaluation scheme
- Speculation!
- Acknowledgements
3How a Quantum Chemist Looks at the World
- An atom or molecule with many electrons, can be
modelled with at least one Slater determinant - Account for Pauli Exclusion Principle and some
- electron-electron repulsion
- Do not treat electron-electron repulsion properly!
In order to account for instantaneous electron
repulsion properly we need to find a basis in
which to perform matrix mechanics
4How a Quantum Chemist Looks at the World
- Solve the time-independent Schrödinger equation
for atomic and molecular systems - Choose a finite one-electron basis set composed
of 2n spin-orbitals. - This lets us write the Hamiltonian in second
quantized form as - For an N-electron system expand exact wave
function in configurations from the totally
antisymmetric tensor product space
5Problems, Problems, Problems!
- This simplistic approach presents a horrendous
computational problem! - The many electron basis scales as
- Three principle approaches to solve the
Schrödinger equation - Configuration Interaction (CI). Requires many
- Perturbation Theory (PT). Requires many hard to
evaluate - Coupled Cluster Theory (CC). Requires hard to
evaluate - Physical inconsistencies creep into the
determinant representation of the many-electron
basis!
6New Tools A Crash Course in Representation
Theory
7(No Transcript)
8A Closer Look At Spin
- Since to good approximation, and the
Hamiltonian for most chemical systems is spin
independent - Also and are invariants in
, so their eigenvalues label irreps of the
su(2)-module of many electron wave functions - Slater determinants are a common and convenient
basis used for many-electron problems (i.e. basis
for ). - Slater dets. are always eigenfunctions of
but not always of ! - This basis yields spin-contaminated solutions
to the Schrödinger eq. - We loose the advantage of partial diagonalization
of in a non-spin-adapted basis.
9Unitary Transformation of Orbitals
- V2n is invariant to unitary transformations
- Through the same analysis
Thus
where
Therefore V2n carries the fundamental irrep,
of U(2n)!
Vn carries the fundamental irrep of U(n)
S2 carries the fundamental irrep of U(2)
10Now For Some Algebra
U(n)
Generators of
U(2)
U(2n)
Lie product of u(n)
11Method to construct eigenstates of
Tensor Irreps of U(n)
- Gelfand and Tsetlin canonical orthonormal basis.
- Gelfand-Tsetlin basis adapted to the subgroup
chain - Irreps of U(k) characterized by highest weight
vectors mk - for N-electron wave functions carries the
totally antisymmetric irrep of U(2n), - Gelfand-Tsetlin (GT) basis of U(2n) is not an
eigenbasis of - We consider the subgroup chain instead
- Only two column irreps of U(n) need to be
considered - The GT basis of U(n) is an eigenbasis of !
12A Detour Through the Strange Land of Para-Fermi
Algebra
- Operators forming a para-fermi algebra satisfy
- The and can be built from the
Green ansatz -
- The and are weird kinds of
Fermions satisfying - However, unitary group generators can be built as
13Para-Fermi Algebra in Quantum Chemistry
- Define the operators
- Then the weird fermions are given as
- And the para-fermi operators are defined through
- Two terms in the sum, therefore electronic
orbitals are para-fermions of order 2
14Clifford Algebra Unitary Group U(2n)
- Consider the multispinor space spanned by
nth-rank tensors of (single particle Fermionic)
spin eigenvectors -
- carries the fundamental reps of SO(m), m
2n or 2n1 and the unitary group U(2n) - carries tensor irreps of U(2n)
- Using para-Fermi algebra, one can show only
of U(2n) contains the p-column irrep of U(n)
at least once. - For the many-electron problem p 2 and thus
- All G2a1b0c of U(n) are contained in G2 of
U(2n), the dynamical group of Quantum Chemistry!
15Where the Clifford Algebra Part Comes in
- The monomials are a basis for the Clifford
algebra Cn - The monomials can be used to construct generators
of U(2n).
primitive Clifford numbers
16- Since m is a vector of 0s and 1s then using
maps - Elements of a 2-column U(n)-module, are
linear combinations of two-box (Weyl)
tableaux
we can go between the binary and base 10 numbers
with m m2
17Action of U(n) Generators on in
Form
- Action of U(2n) generators on is
trivial to evaluate - Since any two-column tableau can be expressed as
a linear combination of two-box tableaux, expand
U(n) generators in terms of U(2n) generators
weights of the ith component in the pth monomial
hard to get sign for specific E
copious!!!
18Action of U(n) Generators on in
Form
- Given a G2a1b0c the highest weight state in
two-box form - Get around long expansion by selecting
out that yield a non-zero result on the
to the right. - Consider with
(lowering generator) - Examine if contains and/or
- e.g. If and
then contains . - Generate r from i and j with p and/or q
- e.g. If contains then
can be lowered to generate the rest of the
module.
19Basis Selection and Generation
- Given a G2a1b0c lower from highest weight state
according to a number of schemes - Clifford-Weyl Basis
- Generate by simple lowering action and thus
spin-adapted - Equivalent to Rumer-Weyl Valence Bond basis
- Can be stored in distinct row table and thus has
directed graph representation - NOT ORTHAGONAL
- Gelfand-Tsetlin Basis
- Generate by Schmidt orthagonalizing CW basis or
lowering with Nagel-Moshinsky lowering operators - Can be stored in DRT
- Orthagonal
- Lacks certain unitary invariance properties
required by open shell coupled cluster theory
20Basis Selection and Generation
- Jezorski-Paldus-Jankowski Basis
- Use U(n) tensor excitation operators adapted to
the chain - Symmetry adaptation accomplished with Wigner
operators from SN group algebra - Resulting operators have a nice hole-particle
interpretation - No need to generate basis explicitly
- Orthagonal and spin-adapted
- Has proper invariance properties required for
open-shell Coupled Cluster - Operators in general contain spectator indices
which lengthen computations and result in even
more unnatural scaling - Determinant Basis
- Just use the two-box tableau
- Easy to generate
- Symmetric Tensor Product between two determinants
- Orthagonal
- NOT SPIN-ADAPTED
21Formulation of Common Correlated Quantum
Chemical Methods
- Equations of all these methods can be formulated
in terms of coefficients (known or unknown)
multiplied by a matrix elements sandwiching
elements of - Configure Interaction
- Coupled Cluster Theory
- Rayleigh-Schrödinger Perturbation Theory
22Formulation of Common Correlated Quantum
Chemical Methods
- One can use induction on the indices of each
orbital subspace - core
- active
- virtual
- This invariance allows one to use numerical
indices on these matrix elements and generate
closed form formulas
to show that the multi-generator matrix elements
are invariant
to the addition or subtraction of orbitals within
each subspace
23Speculation
- A composite particle composed of two electrons
occupy geminals - There are various schemes to form geminals from
pairs of orbitals - Most forms of geminals behave as bosons form a
second-quantized point of view - The geminals that are formed from bosonized
orbitals are really only approximate bosons, form
a purists point of view - Also, there are lots of ways to bosonize orbitals
- An openshell molecule is composed of paired and
unpaired electrons in orbitals - SPECULATION! Can we treat general openshell atoms
and molecules as interacting systems of
(approximate) bosons and fermions?
24Supersymmetry in Quantum Chemistry?
- Composite systems of interacting bosons and
fermions with the added symmetry that boson and
fermions can transform into one another can be
treated be Lie superalgebras - In nuclear physics there is the supersymmetric
interacting boson-fermion models of Iachello et
al. - Pairs of nucleons are treated approximately as
bosons - Any unpaired nucleons are treated as fermions
- Excitation of a boson to a fermion changes the
net particle number and interrelates nuclei of
different masses by classifying them as
transforming as the same superirrep (the totally
supersymmetric one) - Model works ok when compared to experiment
- Can we interrelate molecules in different sectors
of Fock space and include proper spin adaptation
by studying U(mn) modules?
25Acknowledgments
- John Jay College and CUNY
- My collaborators and colleagues
- Prof. Josef Paldus
- Prof. Marcel Nooijen
- Prof. Debashis Mukherjee
- Sunita Ramsarran
- Chris Barden
- Prof. Jon Riensrta-Kiracofe