Download Easy Spin-Symmetry-Adaptation. Exploiting the Clifford

“Easy” spin symmetry adaptation
Exploitation of the Clifford Algebra Unitary Group in
Correlated Many-Electron Theories
U ( 2n )
U (n)
 Chemistry!
Nicholas D. K. Petraco
John Jay College and the Graduate Center
City University of New York
Outline
●
●
Quantum Chemistry and many-electron wave functions
Solving the Schrödinger equation including electron
correlation
●
Spin-adaptation and some algebra
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Representation theory of the unitary group
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The Clifford algebra unitary group
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U(n) module in U(2n) form
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Matrix element evaluation scheme
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Acknowledgements
How a Quantum Chemist Looks at the World
●
An atom or molecule with many electrons, can be modelled with
at least one Slater determinant
Orbital
Energy
N
...
2
1
orbitals
–
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
t u
t u
r s
r s
c d
c d
a b
a b
st
bc
HF
How a Quantum Chemist Looks at the World
•
Solve the time-independent Schrödinger equation for atomic
and molecular systems
o
Choose a finite one-electron basis set composed of 2n spinorbitals.
o
This lets us write the Hamiltonian in second quantized form as:
o
For an N-electron system expand exact wave function in
“configurations” from the totally antisymmetric tensor product
space:
Problems, Problems, Problems!
This simplistic approach presents a horrendous computational
problem!
●
The many electron basis scales as:
o
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Three principle approaches to solve the Schrödinger equation
1.
Configuration Interaction (CI)
2.
Perturbation Theory (PT)
3.
Coupled Cluster Theory (CC)
o
o
o
●
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!
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
adapted basis.
in a non-spin-
Unitary Transformation of Orbitals
●
V2n is invariant to unitary transformations:
Thus:
where
Therefore V2n carries the fundamental irrep,
●
of U(2n)!
Through the same analysis:
Vn carries the fundamental irrep of U(n)
S2 carries the fundamental irrep of U(2)
Now For Some Algebra
●
Let
Generators of: U(2n)
and with:
U(n)
Lie product of u(n):
U(2)
Methods to construct eigenstates of
●
●
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 Wick’s theorem to
the resulting matrix elements

Straight forward but algebra messy and auto-programs (tensorcontraction-engines hard to come by)
Approach 3: Tensor Irreps of U(n)
Gel’fand and Tsetlin formulated the canonical orthonormal basis
for unitary groups.
●
●
o
Gel’fand-Tsetlin basis adapted to the subgroup chain:
o
Irreps of U(k) characterized by highest weight vectors mk
o
Irreps are enumerated by all partitions of k
o
Partitions conveniently displayed as Young tableaux (frames)
for N-electron wave functions carries the totally
antisymmetric irrep of U(2n),
●
Gel’fand-Tsetlin (GT) basis of U(2n) is not an eigenbasis of
●
We consider the subgroup chain instead:
Tensor Irreps of U(n)
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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
irrep.
is at most a two row irrep,
is at most a two column
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
!
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 2n+1 and
the unitary group U(2n)
●
carries tensor irreps of U(2n)
o
●
Using para-Fermi algebra, one can show only
contains the p-column irrep of U(n) at least once.
of U(2n)

For the many-electron problem take p = 2 and thus

All G[2a1b0c] of U(n) are contained in G[2] of U(2n), the dynamical group
of Quantum Chemistry!
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).
●
Since m is a vector of 0’s and 1’s then using maps:
we can go between the binary and base 10 numbers
with m = m2
●
Elements of a 2-column U(n)-module,
are linear
combinations of
two-box (Weyl) tableaux
Action of U(n) Generators on
●
●
Action of U(2n) generators on
in
Form
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!!!
Action of U(n) Generators on
in
Form
Given a G[2a1b0c] the highest weight state in two-box form
●
can be lowered to generate the rest of the module.
Get around long
expansion by “selecting out”
a non-zero result on the
to the right.
●
–
Consider
●
with
Examine if
e.g. If
●
contains
and
(lowering generator)
and/or
then
Generate r from i and j with p and/or q
e.g. If
contains
then:
that yield
contains
.
Action of U(n) Generators on
–
Sign algorithm for non vanishing
in
Form
:
●
Convert indices of
to digital form.
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“Bit-wise" compare the two weight vectors,
●
Sign is computed as (-1)#open pairs
and
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
Basis Selection and Generation
●
Given a G[2a1b0c] 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
Gel’fand-Tsetlin Basis

Generate by Schmidt orthagonalizing CW basis or lowering with NagelMoshinsky lowering operators

Can be stored in DRT

Orthagonal

Lacks certain unitary invariance properties required by open shell coupled
cluster theory
Basis Selection and Generation

Jezorski-Paldus-Jankowski Basis

Use U(n) tensor “excitation” operators adapted to the chain:


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


Symmetry adaptation accomplished with Wigner operators from SN group algebra
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
Formulation of Common Correlated
Quantum Chemical Methods
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Equations of all these methods can be formulated in terms of
coefficients (known or unknown) multiplied by a matrix
elements sandwiching elements of U[u(n)]
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Configure Interaction
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Coupled Cluster Theory
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Rayleigh-Schrödinger Perturbation Theory
Formulation of Common Correlated
Quantum Chemical Methods
One can use induction on the indices of each orbital subspace
●
–
core
–
active
–
virtual
to show that the multi-generator matrix elements are invariant
to the addition or subtraction of orbitals within each subspace
–
The invariant allows one to use numerical indices on these matrix
elements and generate closed form formulas
e.g. Consider the Coupled Cluster term:
Evaluate
and
To get a closed form matrix element we only need to evaluate
Only evaluate
and
Acknowledgments
●
Sultan, Joe and Bogdan
●
John Jay College and CUNY
●
My collaborators and colleagues:
o
Prof. Josef Paldus
o
Prof. Marcel Nooijen
o
Prof. Debashis Mukherjee
o
Sunita Ramsarran
o
Chris Barden
o
Prof. Jon Riensrta-Kiracofe