Pair-number density-functional theorybased on multi-configurational

self-consistent field method

A. HolasInstitute of Physical Chemistry of the Polish Academy of Sciences,

Kasprzaka 44/52, 01–224 Warszawa, Poland

Lecture presented atNeutron Scattering Spectroscopy and Related Problems

27th Janik’s Friends MeetingZakopane, 10–16 July, 2011

OUTLOOK

● Importance of DFT and DM-FT approaches to Quantum Chemistry

● Basic definitions and notation

● Energy terms as functionals of reduced DMs

● Advantages and shortcomings of DFT and Pair-DFT

● Basics of Pair-DFT

●Multi-configurational (MC) self-consistent field (SCF) method

● Pair-DFT in the MC space appropriate for given molecule (PDFT-MC)

● Algorithm to solve the GS problem using PDFT-MC

● Properties of and approximations for correlation energy functional

● Determination of correlation-energy coefficient using virial equation

● Conclusions (simple test, static and dynamic correlation and others)

● Density-functional theory (DFT)Density-matrix-functional theory (DM-FT)

Importance of these approaches to Quantum Chemistry ●

DFT — working horse of modern computational chemistry and physics.

Many-electron systems: atoms, molecules (including very large), clusters, nanostructures, solids (bulk, surface, molecule at surface, …), …,

sufficient accuracy for many properties,computational affordability.

DM-FT — alternative approaches (under developing) to overcome difficulties of DFT.

All aimed to make possible calculations for larger and larger systems with “chemical accuracy”

● Basic definitions and notation ●

N-electron system operators (atomic units used)

For actual system (atom, molecule, cluster) in Born-Oppenheimer approximation:

— N–electron Hilbert space of antisymmetric wave functions.nondegenerate ground state (GS)

assumed

contraction of pure-state N–DM

● Energy terms as functionals of reduced DMs ●GS solution:

2-DM

antisymmetric for

spinless 2-DM

diagonal of spinless 2-DM ≡ spinless 2-density ≡ pair-number density ≡ pair density

1-DM

spinless 1-DM

diagonal of spinless 1-DM ≡ spinless 1-density ≡ particle-number density ≡ density

note

hermitian

Usually is denoted as .

as functionals of all energy terms are exact,

Summary:

as functional of

as functional of

as functional of

needs approximation,

needs approximation,

needs approximation,

The GS energy — from minimization of the total energy functional with respect to DM

Obstacle: N-representability condition (arising from the Fermi statistics of electrons) needs to be imposed on DM.

Search for these conditions: Coleman — since 1951 till now,Valdemoro, Nakatsuji, Mazziotti — recent works.

Relevant books:

■ Reduced Density Matrices: Coulson's Challenge.A.J. Coleman, V.I. Yukalov. Lecture Notes in Chemistry Vol. 72. (Springer, Berlin 2000).

■ Reduced-Density-Matrix Mechanics. With Application to Many-Electron Atomsand Molecules. D.A. Mazziotti,I. Prigogine, S.A. Rice (eds.). Advances in Chemical Physics, Volume 134 (Wiley-Interscience, Hoboken, 2007).

Conditions known for and for . Large number of necessary conditions is knownfor other DMs, but their implementation prohibitively tedious and expensive.

● Advantages and shortcomings of DFT and Pair-DFT ●

DFT: independent functional variable — the particle-number density ,(1-density)to be N–representable:

needs approximation, while is known exactly.

DFT-KS (DFT based on the Kohn-Sham approach): provides , so partition:

it needs approximation now!

Many reasonable approximations are known, however usually plagued by by the “self-interaction error”

PDFT (Pair-DFT): independent functional variable — the pair-number density(2-density)N-representability conditions for are unknown.

Only needs approximation, while and are known exactly (so, no “self-interaction error”).

PDFT-MC (PDFT based on Multi-Configurational SCF method): provides N-representable , and , thus allowing for partition:

the correlation kinetic energy: it needs approximation now!

● Basics of PDFT ●

note

Variational principle for GS:

Space of N-representable 2-densities:

total effective pair interactionMinimization in two steps:

note:

KEF (kinetic-energy functional) definition, according to Levy's “constrained search”[Ziesche (1994), Levy and Ziesche (2001)]

■ Hohenberg-Kohn theorems for PDFT ■

(Henderson, 1974, Ziesche, 1996)

First:

or, equivalently, for given N

Second:with the definition

Note: is the GS 2-density,

● MCSCF method ●one of standard models of quantum chemistry.

MC ≡ multi-configurational, SCF ≡ self-consistent field.

For details see, e.g., M.W. Schmidt and M.S. Gordon, Annu. Rev. Phys. Chem. 49, 233 (1998).T. Helgaker, P. Jorgensen, and J. Olsen, Electronic Structure Theory

(Wiley, Chicester, 2000).

configuration ≡ single Slater determinant (SD) wave function constructed of a chosen set of orthonormal spin orbitals

SC-SCF (single-configurational SCF) method ≡ Hartree-Fock (HF) methodSD space — a subspace of N-electron Hilbert space defined as

(variational freedom — a set of orthonormal spin orbitals)

HF energy

Necessary conditions for minimum, with Lagrange multipliers

are equivalent to HF equations

iterative solving until SCF solution:

Steps towards MC wave function.

From the N self-consistent-field orbitals obtain

and then solve

Criteria for choice of methods

� when is finite and large

— single-configurational SCF method is sufficient;

— multi-configurational SCF method [using spin orbitals] is needed, because

� when for , but is finite and large,

where are constructed of N-element subsets of

N.b., it occurs for molecular geometries far from equilibrium

Slater determinants entering the MC wave function

Given molecular orbitals ( occ = occupied, i.e., used in determinants)

The molecular spin orbitals are

The set of all orbitals is divided into two subsets,

inactive orbitals occurring in all determinants as doubly occupied,

with the corresponding subset of spin orbitals

active orbitals yielding the subset of spin orbitals

Each Slater determinant is constructed of all inactive spin orbitals and of

active spin orbitals chosen from ones. Note

The maximum number of possible configurations (cnf) is

The actual number of configurations can be chosen from the range

according to some criteria, e.g., of spin and/or spatialsymmetry, etc.

Special cases of MC wave functions:

for — Hartree-Fock (HF) wave function

for — complete active space (CAS) wave function

The MC variational (trial) wave function

Variational parameters of MC function:

— MC wave function expansion coefficients, constrained by

— molecular orbitals, constrained by

In practice, orbitals are expanded in some basis, ,appropriate for the given molecule and to the desired accuracy, so

Then alternative variational parameters are

— orbital expansion coefficients, constrained by

Full information describing the MC trial function

MC space — a subspace of N-electron Hilbert space,appropriate for the given molecule, defined as

Variational freedom in due to parameters and

The MC SCF solution

Minimization is performed iteratively, by applying:

(i) rotations to the initial orthonormal

(ii) norm-conserving corrections to the initial normalized

until selfconsistent values of all parameters are reached.

(i.e., GS energy of system in space)

Correlation energy from estimates of the GS energy

because

(full correlation energy) (static correlation energy) (dynamic correlation energy)

● PDFT in the MC space appropriate for the given molecule ●

� MC space of pair densities

Obviously, these pair densities are N-representable.

� The MC-space kinetic-energy functional (MC-KEF)

Algorithm for its explicit calculation is available.

� The correlation KEF of MC-PDFT (its meaning — the dynamic correlation energy)

should be approximated for practical applications by some explicit functional

(similarly as of DFT is approximated, e.g.,

etc.)

� GS problem reformulated using MC-KEF.

The GS energy and 2-density obtained from

The necessary condition for the minimum, provided is

— correlation pair potential.

Calculational scheme stemming from Eq. (A) will be formulated after

discussion about

(A)

Here

i.e.,

� Reference system in MC space

The original system (molecule) is characterized by

where

Consider a reference system (RS) characterized by

Its GS 2-density is a unique functional of i.e., it satisfies

where will be close to

Perform MC SCF calculations for the RS(i.e., solve the GS problem of the RS in the space,

nondegenerate solution is assumed)

Two-step minimization

Definition of MC-space KEF was applied

The minimizer is a unique functional of i.e., it satisfies

The necessary condition for the minimum, provided is

(B)

● Algorithm to solve the GS problem using PDFT-MC ●

Rewrite Eq. (B) concerning the RS

and compare it with Eq. (A) concerning the original system

Two above equations can be equivalent provided

leading to

● Note: in DFT, the set of noninteracting v-representable 1-densities is densein the set of interacting v-representable 1-densities, and vice versa.

This means:

For every for arbitrary from one set, we can find from the second set, such that

(R. van Leeuwen, 2003).

We expect the same property holds in PDFT for u-representable 2-densities:

and

So, in practice, we will not distinguish from

● End note

The established equivalence relation allows to reformulate the GS problem as two subproblems:

� Finding the GS 2-density

Note: this is standard MC SCF problem, but with the modified pair interaction potential

— by performing the MC SCF calculations with the modified Hamiltonian

As initial take the minimizer of the original MC SCF problem, performminimization applying MC SCF methods.

� Finding the GS energy

Note:

● Exact properties of correlation KEF

� Negativity: because

� Scaling: where

● Approximation of correlation KEF ●Model functionals satisfying scaling and negativity properties (Levy and Ziesche, 2001):

where — real function of two real arguments. Corresponding pair potential:

� Type A: — integer

� Type B:

— integer; — real constants; — some real function of a real argument;

example:

example:

Note: linear combination (with positive coefficients) of these approximations also satisfies the same properties.

● Determination of from virial theorem (equation) ●

, which is valid

for atHere

Note: for an atom

Steps

Demand virial equation, with

inserted, to be satisfied when the MC SCF solution for is reached.

(i) Choose initial weak to have(i.e., correlation KEF as a small correction to MC KEF)

(ii) Perform MC SCF calculations for system with

(iii) Find from virial equation at and

During iterations the set of parameters is improved.

● Advanced approximation of ●

Proposed steps towards construction: for a few atoms and molecules needing in CM SCF approach

� test some terms of type-A approximation,� test some functions and terms of type-B approximation,� for final choose a linear combination of well performing terms

Note: is determined from virial equation, are to be universal.

� determine parameters by fitting the calculated properties to a database of molecular properties (for a "training set" of molecules). Such fitting is already used in the case of DFT semiempirical approximations to

● Test result of PDFT-SC ●(i.e., using only the single-configuration (SC) space, instead of MC space)

performed by Higuchi and Higuchi, Phys. Rev. B 78, 125101, 2008, for neutral neon atom, using one-term type-A approximation of

However, the constant is not determined from the virial equation, but from best fit of their 1-density (obtained by PDFT-SC with fixed K of )to the 1-density (taken from the high-accuracy configuration-interaction (CI)calculations of Bunge and Esquivel, Phys. Rev. A 34, 853,1986).

The best K is the minimizer of the root-mean-square error (RMSE):

radia

l 1-d

ensi

ty

distance from nucleus

Higuchi and HiguchiNe

The best fit of 1-density

This test covers about 20% of the exact electron correlation energy = Etotal – EHF,

despite so simple, one-parameter approximation to the correlation energy functional

Higuchi and Higuchi

K — the parameter of the correlation energy

● Conclusions ●

■ Due to constructing from the MC wave function, the pair density is obviously N-representable.

■ Due to the exact expression for the e-e interaction energy in PDFT in terms of the pair density, there is no self-interaction error that plagues DFT and (1-DM)FT.

■ The test result shows that PDFT is a plausible calculational tool, improvingsignificantly the HF results even in the case of a very primitive approximation for the correlation energy and the simplest version PDFT-SC rather than PDFT-MC.

■ When PDFT-MC is used, the static correlation is fully included, while the correlationkinetic energy (approximated by model functional) represents the dynamic correlation.

■ Approximation for the correlation kinetic energy can be improved by using multi-parameter expressions satisfying exactly the scaling property.

■ The expectation value of an arbitrary two-particle operator can be evaluatedrigorously using the pair density of PDFT [it is not available in DFT and (1-DM)FT].