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Easy spin symmetry adaptation
Exploitation of the Clifford Algebra Unitary
Group 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

• Quantum Chemistry and many-electron wave
  functions
• Solving the Schrödinger equation including
  electron correlation
• Spin-adaptation and some algebra
• Representation theory of the unitary group
• The Clifford algebra unitary group
• U(n) module in U(2n) form
• Matrix element evaluation scheme
• 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
• Consist of atomic orbitals and fitting
  coefficients, molecular orbitals (MOs)
• Account for Pauli Exclusion Principle
• Do not treat electron-electron repulsion properly!

To account for instantaneous electron correlation
properly we need to form linear combinations of
excited dets from a suitable reference
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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)
• Perturbation Theory (PT)
• Coupled Cluster Theory (CC)
• CI can be formulated in the entire many-electron
  basis (FCI) or truncated (CISD, CISDT, etc.)
• PT and CC must be evaluated in a truncated
  many-electron basis (MP2, MP3, etc. or CCSD,
  CCSDT, EOM-CCSD, etc.)
• Despite basis truncation scaling is still rather
  terrible
• Physical inconsistencies creep into the
  determinant representation of the many-electron
  basis!

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A Closer Look At Spin

• To good approximation, the Hamiltonian for most
  chemical systems is spin independent
• Thus and
• The (tensor product) basis for our
  spin-independent Hamiltonian can be written as a
  direct sum of invariant subspaces labeled by
  eigenvalues of and
• 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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• Approach 1 Use SU(2) single particle spin
  coupling techniques and perhaps graphical methods
  of spin-algebras (Jucys diagrams)
• No democratic way to couple odd numbers of
  particles.
• Orbital to spin-diagram translation error prone
  diagram algebraic translating
• Automatic implementation???
• Approach 2 Spin-adapt normal ordered excitation
  operators using SN group algebra elements and
  apply Wicks theorem to the resulting matrix
  elements
• Straight forward but algebra messy and
  auto-programs (tensor-contraction-engines hard to
  come by)

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Approach 3 Tensor Irreps of U(n)

• Gelfand and Tsetlin formulated the canonical
  orthonormal basis for unitary groups.
• Gelfand-Tsetlin basis adapted to the subgroup
  chain
• Irreps of U(k) characterized by highest weight
  vectors mk
• Irreps are enumerated by all partitions of k
• Partitions conveniently displayed as Young
  tableaux (frames)
• 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

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Tensor Irreps of U(n)

• However we must consider the subduction
• Noting that
• By the Littlewood-Richardson rules is
  contained only once in if the
  irreps in the direct product are conjugate.
• Since is at most a two row irrep,
  is at most a two column irrep.
• Thus the only irreps that need to be considered
  in the subduction are two column irreps of the
  (spatial) orbital unitary group U(n)
• The GT basis of U(n) is an eigenbasis of !

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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 take 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 and
Other Trivia

• The monomials are a basis for the Clifford
  algebra Cn
• The monomials can be used to construct generators
  of U(2n).

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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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Action of U(n) Generators on in
Form

• Sign algorithm for non vanishing
• Convert indices of to digital form.
• Bit-wise" compare the two weight vectors,
  and
• Sign is computed as (-1)open pairs
• An open pair is a "degenerate" (1,1) pair of
  electrons above the first (1,0) or (0,1)
  pair.
• e.g. If (1 1 1 1 1 0 0 0 1 0 1 1
  0 0 1)
• (1 1 1 1 1 1 0 0 1 0 0
  0 1 0 1)

then sign -12 1
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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 Uu(n)
• 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
• The invariant 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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e.g. Consider the Coupled Cluster term
Evaluate
and
To get a closed form matrix element we only need
to evaluate
and
Only evaluate
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Acknowledgments

• Sultan, Joe and Bogdan
• 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