Algebraic Symmetries in Quantum Chemistry

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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

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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

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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
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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

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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!

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New Tools A Crash Course in Representation
Theory
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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.

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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)
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Now For Some Algebra

• Let and with

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

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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

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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

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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!

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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
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• 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
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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!!!
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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.
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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

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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

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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

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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
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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?

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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?

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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