Download Algebraic Symmetries in Quantum Chemistry

Algebraic Symmetries in Quantum
Chemistry
Clifford Algebra and Para-Fermi Algebra 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
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Part I, The Setting
–
Quantum Chemistry and many-electron wave functions
–
Solving the Schrödinger equation including electron correlation
Part II, Mathematical Tools
–
●
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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
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
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
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
●
●
Three principle approaches to solve the Schrödinger
equation
1.
Configuration Interaction (CI). Requires many
2.
Perturbation Theory (PT). Requires many hard to evaluate
3.
Coupled Cluster Theory (CC). Requires hard to evaluate
Physical inconsistencies creep into the determinant
representation of the many-electron basis!
“New Tools”
A Crash Course in Representation Theory
●
●
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A “group” is a special collection of “operators” which transform a given
set of “vectors” V, among themselves

Really we only need the “algebra” associated with the group

Our vectors will be many-electron wave functions

The operators that make up our algebra are electron excitation operators and
electron number operators
A “representation” G is a recipe (homomorphism) for writing each
operator in the algebra A as a “matrix”
Given a “vector space”, V and an algebra A then
subspace” (“invariant module”) if
is an “invariant
An “irreducible module” (“irreducible representation space” or
“irrep”) is an invariant subspace that cannot be written as a sum of
smaller invariant subspaces
●
Every irrep can be labelled by a list of “weights”, G[m]
●
o
o
o
o
Our lists of weights correspond to pictures called “Weyl frames”
Weights are eigenvalues of “invariance operators”
algebra
, made from elements in the
Invariance operators are in general elements in the “universal enveloping algebra”
of A,
Every vector in the irrep is assigned a unique pattern of weights stemming from the
algebra A and any if its subalgebras
●
The weight patterns we will use are called “Weyl tableaux”
A Closer Look At Spin
●
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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
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)
Method to construct eigenstates of
Tensor Irreps of U(n)
Gel’fand and Tsetlin canonical orthonormal basis.
●
o
Gel’fand-Tsetlin basis adapted to the subgroup chain:
o
Irreps of U(k) characterized by highest weight vectors mk
o
for N-electron wave functions carries the totally antisymmetric irrep
of U(2n),
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Gel’fand-Tsetlin (GT) basis of U(2n) is not an eigenbasis of
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We consider the subgroup chain instead:
–
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Only two column irreps of U(n) need to be considered
The GT basis of U(n) is an eigenbasis of
!
A Detour Through the Strange Land of
Para-Fermi Algebra
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Operators forming a para-fermi algebra satisfy
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The
o
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The
and
and
can be built from the “Green ansatz”
are “weird” kinds of Fermions satisfying
However, unitary group generators can be built as
Para-Fermi Algebra in Quantum
Chemistry
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Define the operators
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Then the “weird” fermions are given as
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And the para-fermi operators are defined through
o
Two terms in the sum, therefore electronic orbitals are para-fermions of
order 2
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)
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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 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
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The monomials are a basis for the Clifford algebra Cn:
primitive “Clifford numbers”
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The monomials can be used to construct generators of U(2n).
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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
●
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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
.
Basis Selection and Generation
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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
:
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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
–
This invariance allows one to use numerical indices on these matrix
elements and generate closed form formulas
Speculation
A composite particle composed of two electrons occupy
“geminals”
●
o
There are various schemes to form geminals from pairs of orbitals
Most forms of geminals behave as bosons form a secondquantized point of view:
●
o
o
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
●
o
SPECULATION! Can we treat general openshell atoms and molecules as
interacting systems of (approximate) bosons and fermions?
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.
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o
Pairs of nucleons are treated approximately as bosons
o
Any unpaired nucleons are treated as fermions
o
o
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“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(m|n)
modules?
Acknowledgments
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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