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Algebraic Symmetries in Quantum Chemistry

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Title: Algebraic Symmetries in Quantum Chemistry


1
Algebraic 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

2
Outline
  • 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

3
How 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
4
How 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

5
Problems, 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!

6
New Tools A Crash Course in Representation
Theory
7
(No Transcript)
8
A 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.

9
Unitary 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)
10
Now For Some Algebra
  • Let and with

U(n)
Generators of
U(2)
U(2n)
Lie product of u(n)
11
Method 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 !

12
A 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

13
Para-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

14
Clifford 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!

15
Where 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
17
Action 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!!!
18
Action 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.
19
Basis 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

20
Basis 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

21
Formulation 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

22
Formulation 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
23
Speculation
  • 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?

24
Supersymmetry 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?

25
Acknowledgments
  • 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
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