“Easy” spin symmetry adaptation

Nicholas D. K. PetracoJohn Jay College and the Graduate Center

City University of New York

Exploitation of the Clifford Algebra Unitary Group in Correlated Many-Electron Theories

!)2()( ChemistrynU

nU

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

How a Quantum Chemist Looks at the World

a

t

c d

sr

u

b a

t

c d

sr

u

bstbcHF

● 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!

...OrbitalEnergy

orbitals1

2

N

To account for instantaneous electron correlation properly we need to form linear combinations of excited dets from a suitable reference

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 spin-orbitals.

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:

● This simplistic approach presents a horrendous computational problem!

o The many electron basis scales as:

● Three principle approaches to solve the Schrödinger equation

1. Configuration Interaction (CI)

2. Perturbation Theory (PT)

3. Coupled Cluster Theory (CC)

o CI can be formulated in the entire many-electron basis (FCI) or truncated (CISD, CISDT, etc.)

o PT and CC must be evaluated in a truncated many-electron basis (MP2, MP3, etc. or CCSD, CCSDT, EOM-CCSD, etc.)

o Despite basis truncation scaling is still rather terrible

● Physical inconsistencies creep into the determinant representation of the many-electron basis!

Problems, Problems, Problems!

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.

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)

2 carries the fundamental irrep of U(2)

Now For Some Algebra

U(n)Generators of: U(2)U(2n)

● Let and with:

Lie product of u(n):

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 (tensor-contraction-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)● 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 !

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)

o 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 [2a1b0c] of U(n) are contained in [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:

● 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

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 componentin the pth monomial

hard to get sign for specific E

copious!!!

Action of U(n) Generators on in Form● Given a [2a1b0c] 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.

– 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)

Action of U(n) Generators on in Form

then sign = -12 = 1

Basis Selection and Generation● Given a [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 Nagel-Moshinsky 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:

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

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 U[u(n)]

● Configure Interaction

● Coupled Cluster Theory

● 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

– The invariant allows one to use numerical indices on these matrix elements and generate closed form formulas

to the addition or subtraction of orbitals within each subspace

to show that the multi-generator matrix elements are invariant

e.g. Consider the Coupled Cluster term:

Evaluate and

To get a closed form matrix element we only need to evaluate

Only evaluate and

● 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

Acknowledgments