Multi-reference Density Functional Theory

COLUMBUS WorkshopArgonne National Laboratory

15 August 2005

Capt Eric V. BeckAir Force Institute of Technology

Department of Engineering Physics2950 Hobson Way, Building 640

Wright Patterson AFB, OH 45433-7765 USA

Overview

• Theory• Hartree-Fock approximation• Kohn-Sham Density Functional Theory (DFT)• Configuration Interaction• DFT augmented CI

• CIUDG-based MR-DFT model• Source of Kohn-Sham orbitals and Vxc

• Off-diagonal loop contribution scaling

The views expressed in this presentation are those of the author, and do not necessarily reflect the views of the United States Air Force or the United States Government

Hartree-Fock Approximation

• Objective is to obtain solution to Schrodinger’s equation

Ψ=Ψ⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇−=Ψ ∑∑∑ ∑

>n

ji iji ia ia

aS E

rRZZ

rZH 1

21

,,

2

βα αβ

βα

• From independent particle model approximation, assume wave function is separable into hydrogenic one-electron products

( ) ( ) ( ) ( )nn rrrRr ϕϕϕ K2211; =Ψ

• Hartree-Fock mean-field theory results from variational optimization of this product wave function to minimize the expectation value of the energy using a single Slater determinant– Forms the basis of many quantum chemistry wave function based approaches– Hartree-Fock energy is accurate to 1st order

• Post Hartree-Fock methods provide better description of electron-electron correlation energy, defined by

FockHartreeexactncorrelatio EEE −−=

Density Functional Theory

• Hohenberg-Kohn theorem, [PRB, 136:864-871 (1964) ]• Electron density uniquely determines the Hamiltonian• Electronic energy is a functional of the electron density, mimized when exact

density is used

• Kohn-Sham equations [PRA 140, 4A, 1133-1138 (1965)]• Based on fictitious non-interacting independent particle model with effective,

local potential• Define local potential such that exact electron density is reproduced• Again, single Slater determinant model*

• Exchange-correlation energy functionals• Universal functional exists, exact form unknown – exact solution to

Schrödinger equation• Approximate functionals developed yield remarkable accuracy, especially with

molecular geometries

• DFT provides accurate, computationally efficient method for inclusion of electron-electron correlation• Extended to excited states via Time Dependent DFT

DFT (cont)

• Exchange-correlation energy functional contains approximations to– Kinetic energy of interacting electrons – Non-local electron exchange energy– Dynamic electron-electron correlation energy

• Kohn-Sham approach success and accuracy rests on the accuracy of approximations to the exchange-correlation functional– Modern functionals are based on numerical fits to Monte-Carlo quantum

mechanical calculations on Fermi gas (Ceperely and Alder, Phys Rev Lett, 45, 7:566-569 (1980)

– Most modern functionals based on generalized gradient approximation to non-local potential, newest meta-GGA functionals include kinetic energy density

• Important properties of Kohn-Sham DFT– K-S orbitals reproduce exact electron density – DENSITY optimal orbitals– Occupied and unoccupied orbitals experience same effective potential– Single Slater determinant model

Electron correlation

• Non-dynamic – failure of single Slater determinant model to accurately describe system• Arises in correlation between states that are highly degenerate, or nearly degenerate• Requires some sort of multiconfiguration multireference method • Both Hartree-Fock and DFT (TDDFT also) suffer from this deficiency, although if exact

universal functional was used, DFT would yield exact solution to non-relativistic Schrodinger equation using only a single determinant

• Dynamic – failure of Hartree-Fock mean-field approximation to adequately describe electron-electron interaction energy• Perturbation theory (MP2, MP4, MBPT) – HF energy typically accounts for a large fraction

of the total energy• Coupled-cluster – popular and accurate, but computationally intensive• Density Functional Theory – approximate correlation density functional• Configuration interaction (CI) – conceptually simple

• Expand wave function in a series of ground and excited one electron wave functions, variationally optimize expansion coefficients

• Including all possible excitations from ground state is known as a Full CI, which is an exact solution to the non-relativistic Schrödinger equation

• Full CI expansions are computationally intensive for all but the smallest molecular systems and basis sets

Configuration Interaction

• Expand wave function in a series of electron configurations representing zero, one, two, …, N excitations from a specified reference state

• First term represents the reference wave function, second term represents all wave functions resulting from a single excitation from the reference state

• Including all N possible excitation from N electron ground state is known as a Full CI expansion– Full CI results in the exact solution to the non-relativistic, many

electron Schrödinger equation

• Slowly convergent

0 0, , , , ,

a a ab ab abc N abc Ni i ij ij ijk N ijk N

a i a b i j abc Nijk N

c c c cΨ = Ψ + Ψ + Ψ + + Ψ∑ ∑ ∑ L LL L

LL

L

DFT augmented CI

• Grimme (CPL 259:128-137(1996) and Grimme and Waletske (JCP 111(13):5645-5655 (1999))

• Method involves using DFT Kohn-Sham orbitals and fictitious Kohn-Sham Hamiltonian as reference Hamiltonian for subsequent CI expansion• DFT provides information about dynamic correlation through

exchange-correlation energy functional – must avoid double counting in this correlation in the subsequent CI calculation

• DFT provides no information on multideterminant nature of the system, use CI calculation to provide information about multi-determinant character

• They called their approached Multireference DFT, however, it was a single reference method

DFT augmented CI (cont.)

• Approximate diagonal elements by

( ) [ ]( )0

ˆ

ˆ

1 | |

exc exc

exc exc

DFT DFT

n nHF KS HF KS

cc cc aa aac a

n n

Ja cexc

w H E w

w H E w F F F F

p aa cc p N ac acn

ω ω

ω ω

− ≈

− − − + − +

−

∑ ∑

∑∑

DFT augmented CI (cont.)

• pJ and p[N0] are empirically determined parameters in the Grimme-Waletske paper• The theoretical value of pJ in TDDFT is 1-xHF, G-W finds

their value in good agreement with the TDDFT value• They found the exchange contribution correction necessary,

and systematically increased with the number of open shells• G-W settled on a linear relationship between the coefficients and

number of open shells

[ ] [ ][ ]

1 10 0

30 0

0p N p N

p N N

α

α

= +

=

DFT augmented CI (cont)

• Off-diagonal elements are approximated in the method by

42

1ˆ ˆwwp EDFTw H w p e w H wω ω ω ω′− ∆′ ′ ′ ′≈

• ∆E is the energy difference between the diagonal integrals of thetwo interacting CSFs

• Energy dependent scaling leaves interactions between nearly degenerate CSFs intact, while rapidly damping out interactions between non-degenerate CSFs

• Avoids double counting of the dynamic correlation, which is included via the correlation functional

DFT augmented CI (cont.)

• Damping ensures double counting of dynamic correlation avoided

• Can dramatically reduce the size of the CI expansion by clever use of the diagonal CSF integrals

Empirical parameters for BHLYP hybrid functional

Multiplicity p1 p2 pJ p[0] α

singlet 0.619 3.27 0.510 0.595 0.106

triplet 0.619 3.27 0.493 - 0.056

CIUDG-based MR-DFT

• Original implementation of G-W method used Turbomol and single reference CI expansions

• CIUDG is a true multi-reference CI code• No longer a DFT augmented CI, the G-W name Multireference DFT can honestly

be applied

• Implementation involves scaling individual loop off-diagonal loop contributions (all-internal, 1-, 2-, 3-, and 4-external loop contributions) just before update of the sigma vector

• Diagonal integrals are cached in memory to facilitate ∆Eww'calculations

• Semi-intelligent segment interaction will be used to improve efficiency

• Spin-orbit capabilities (unscaled) make MR-DFT attractive for heavy element systems

CIUDG based MR-DFT

• Only one problem, where does one get Kohn-Sham orbitals?

• Multiple approaches here• Modification of SCFPQ to facilitate numerical integration of

exchange-correlation energy functionals• Fairly straightfoward to code• Substatial time investment, with debugging and validation testing

• Interface COLUMBUS to external program source of exchange-correlation integrals and K-S molecular orbital coefficients from a converged DFT (restricted) calculation

NWChem Interface

• NWChem 4.7 provides restricted and unrestricted DFT implementations with many exchange-correlation functionals

• Modification of NWChem 4.7 code to allow extraction of Vxc matrix from converged DFT calculation

• Wrote conversion program that currently takes these xcintegrals, and adds them to the electron-nuclear potential energy integrals• Good news -- this quick and dirty approach reproduces the

NWChem DFT K-S orbital energies and coefficients within SCFPQ

• Bad news -- NWChem Total DFT energy doesn't match SCFPQ total energy

NWChem interface (cont)

• Why the total energy mismatch?• Vxc energy needs to be contracted with the two-electron density,

not the one electron density

• Solution is semi-straightfoward, but not entirely trivial and requires modification of SCFPQ• Create new one-electron integral type within SIFS• If DFT calculation flag set in SCFPQ, add Vxc to Ppqrs matrix

• This doesn't effect the orbital energies of MOs, but is necessary in order to get the SCFPQ total energy to match the NWChem DFT total energy

MR-DFT via CIUDG

• Once Vxc integrals and Kohn-Sham molecular orbital coefficients are converted from NWChem, proceed with CIUDG procedure as normal• CIDRT to define the reference space• TRAN to transform the AOs to MOs based on aoints, aoints2,

and mocoef file• CISRT to sort the various mo integrals into all-internal, 1-, 2-, 3-

, and 4-external integrals• MR-DFT is enabled within COLUMBUS version 6 CIUDG via

new namelist control variables (set to false by default)

MR-DFT specific CIUDG Namelist Input

• p1 = 0.614• p2 = 3.27• od_allint_damp = .true.• od_1ext_damp = .true.• od_2ext_damp = .true.• od_3ext_damp = .true.• od_4ext_damp = .true• dump_diag = .true.• debug = .true.

Summary

• Theory• Hartree-Fock approximation• Kohn-Sham Density Functional Theory (DFT)• DFT augmented CI method of Grimme and Walteske (MR-

DFT)• CIUDG-based MR-DFT in COLUMBUS

• Source of Kohn-Sham molecular orbital coefficients and exchange-correlation integrals from a converged DFT calculation in progress

• Off-diagonal damping code in place• Validation and testing of modifications soon• Efficiency optimization and semi-intelligent segementation

interaction based on CSF diagonal integrals analysis

Questions?