Orbital dependent correlation potentials
in ab initio density functional theory
noniterative - one step - calculations
Ireneusz Grabowski
Institute of Physics
Nicolaus Copernicus University
Toruń, Poland
OEP Workshop, Berlin March 2005
Collaborators:
•
•
•
•
•
So Hirata
Stanislav Ivanov
Victor Lotrich
Igor Schweigert
Rod Bartlett
Quantum Theory Project
University of Florida,
Gainesville, FL
Two independent theories of the electronic structure of atoms, molecules and solids.
WFT DFT
• Its one particle structure • Expensive, but provides make it possible to treat results which are much larger systems than quaranteed to converge to WFT.
the solution of the Schrödinger equation as • Almost all unknown electron correlation and informations are contained basis set is extended. in an exchange –correlation functional Exc and its associated potential Vxc.
• MBPT(2) < CCD < CCSD <
CCSD(T) < CCSDTQ < FCI
• In standard formulation ‐
parameter dependent
• Whereas WFT is a „constructive” theory that provides a prescription for obtaining increasingly
more accurate solutions of Schrödinger equation,
DFT provides the existence of Exc, but does not
provide the energy functional (or even theoretical
prescription) , nor systematic converging series of
approximations to it.
• Standard LDA, GGA, or hybrid functionals works
well in some cases but usually results are
unpredictable and it is difficult (or impossible) to
improve the functional approximation
systematically.
Method which could define a rigorious exchangecorrelation functional, potential and orbitals in context of
the Kohn-Sham theory:
The exploiting in DFT orbital-dependent
functionals and potentials – OEP method.
Ab initio density functional theory
• From coupled-cluster theory and many-body
perturbation theory we derived the local exchangecorrelation potential of DFT in an orbital dependent
form.
• Parameter free
• It guarantees to converge to the right answer in the
correlation and basis set limit, just as does ab initio
WFT.
• Specyfying initially to second-order terms –
Optimized Effective Potential Method with correlation
included - OEP-MBPT(2)-KS, OEP-MBPT(2)-f,...
Two different ways to obtain OEP methods
with correlation included
• Functional derivative path •
– taking functional derivative
with respect to density of
orbital dependent energy
functional (from ab initio WFT) to get exchange‐
correlation potential in KS theory. ƒ MBPT(2) level
ƒ Problems with extending to higher orders
Density condition path
General theoretical
framework based on the
density condition in Kohn
Sham theory involving
coupled cluster method, many body perturbation
theory, and technically
diagramatic manipulation.
Basic formalism – functional derivative path
The spin densities ρσ (r) and KS orbitals {φpσ(r)} are obtained by
a self consistently solving the KS equation:
The local exchange-correlation potential is formally defined as
the functional derivative of the exchange-correlation energy
In the Optimized Effective Potential (OEP) method,
the ExcOEP[{φpσ}] is an explicit functional of spinorbitals,
and the spinorbitals are the solutions of the KS equation
with a local effective OEP potential, which is determined
by the condition that its orbitals be ones that minimize
the energy functional:
The resulting integral OEP equation have to be solved for
the VXC in each KS(OEP) iteration,
Formally we can represent Vxc as
Where Xsσ-1 is the inverse of the static KS linear response
function of a system of noninteracting particles
In the LCAO-OEP procedure, the potential and response
function and it inverse are represented in the AO basis.
Exc=Ex+Ec
Ex - HF exchange energy functional in terms of KS orbitals
Vx – orbital dependent exchage-only OEP potential
Orbital dependent OEP–DFT correlation functional
Energy expression for the second order RS Perturbation
Theory
OEP-MBPT(2)-KS method
I. Grabowski, S. Hirata, S. Ivanov, R. J. Bartlett Ab-initio density functional theory: OEPMBPT(2) – a new orbital-dependent correlation functional.,J. Chem. Phys. 116, 4415 (2002
Density condition in KS theory & Coupled Cluster theory
The KS density by construction is an exact density,
then any corrections to the converged KS density
introduced by changes in φi(r) have to vanish.
ρ(r)= ρKS(r)+δρ(r)
and
δρ(r)=0
The total density from Coupled Cluster (CC) density matrix
We can represent CC density using antisymmetrized diagrams
Equivalence with the
OEP-MBPT(2)
correlation potential
derived from
functional derivative
path.
17
For defining our perturbation at the second order level, we have several different choices for the partitioning of the
Hamiltonian.
• OEP‐MBPT(2)‐KS
H0 =
+
ε
{
p
∑ p p}
p
• OEP‐MBPT(2)‐f
H0 =
∑
f pp { p + p}
p
• OEP‐MBPT(2)‐sc
H0 =
∑
p
f pp { p + p} + ∑ f ij {i + j} + ∑ f ab {a + b}
i≠ j
a ≠b
19
Results
• We are NOT doing CC calculations or more
complicated MBPT(2) !
• We are doing KS DFT‐OEP iterations with
correctly defined exchange‐correlation potentials
(orbital dependent)
• In each KS DFT iteration, using one‐ and two‐
electron integrals we calculate VXC,
• Going back with VXC to the KS equation we obtain new set of orbitals and then we can repeat
our procedure until self consistency is achieved.
Correlation potential of helium
0,04
C o rre la t io n p o t e n t ia l / a .u .
0,02
0,00
0
1
2
3
4
5
6
7
-0,02
Exact (Umrigar & Gonze)
-0,04
Vosko-Wilk-Nusair correlation potential
-0,06
Lee-Yang-Parr correlation potential
KLICS correlation potential
-0,08
OEP-MBPT(2)-KS
-0,10
R / a.u.
OEP-MBPT(2)-f
21
22
Correlation potential of neon (Roos-ATZPU basis set)
0,32
VWN
LYP
0,27
Correlation potential / a.u.
exact correlation potential
0,22
OEP-MBPT(2)SD and D
OEP-MBPT(2)-SD-f
0,17
0,12
0,07
radial charge density
0,02
-0,03 0
1
2
3
4
-0,08
-0,13
R / a.u.
5
6
7
Exchange-correlation potential of neon (Roos-ATZPU)
one step calculations
0
0
1
2
3
4
5
-1
-2
Potential / a.u.
-3
-4
-5
-6
exchange-correlation OEP-MBPt(2)-f and
OEP-MBPT(2)
`exchange-only' OEP
SVWN
BLYP
-7
OEP-MBPT(2)-f-1shot-HF
-8
-9
R / a.u.
6
Correlation potentials of Be atom
0,10
0,00
0
1
2
3
4
5
6
Correlation potential / a.u.
-0,10
-0,20
OEP-MBPT(2)-f
vc exact
-0,30
OEP-MBPT(2)-KS - (non converged)
LYP
VWN
-0,40
-0,50
R / a.u.
7
Energy surface of He2
(17s10p2d)
0,002
E-E∞ / eV
0,001
3
4
5
6
7
8
9
10
11
-0,001
MBPT(2)
-0,002
CCSDT
OEP-MBPT(2)
-0,003
PBE
-0,004
r / au
12
Ne2 dimer potential energy - AUG-CC-PVTZ basis set
300
250
MP2
200
CCSD
CCSD(T)
150
Energy[cm -1 ]
OEP_MBPT(2)-fsc
100
50
0
5
5,5
6
6,5
-50
-100
-150
-200
r [a.u]
7
7,5
8
Approximated one step calculations
• Using exchange-only OEP orbitals we can generate
correlation potentials using one step procedure by
simply inserting orbitals into an orbital-dependent
expression for the correlation potential.
• We can even do the same one step procedure using
HF orbitals, and then generate correlation and
exchange correlation potential, without doing any
OEP & KS self interaction procedure.
Correlation potential of helium 1 step calculations
0,04
C o rrelatio n p o ten tial / a.u .
0,02
0,00
0
1
2
3
4
5
6
7
-0,02
-0,04
-0,06
-0,08
-0,10
R / a.u.
Exact (Umrigar & Gonze)
Vosko-Wilk-Nusair correlation potential
Lee-Yang-Parr correlation potential
KLICS correlation potential
OEP-MBPT(2)SD-f
OEP-MBPT(2)-f-SD-1 shot
OEP-MBPT(2)-SD-f-1shot-HF
Correlation potential / a.u.
Correlation potential of neon 1 step calculations
0,32
VWN
0,27
LYP
exact correlation potential
0,22
OEP-MBPT(2)-f-SD and D
OEP-MBPT(2) -f-SD 1 shot
0,17
OEP-MBPT(2)-f-SD-1 shot HF
0,12
0,07
0,02
-0,03 0
1
2
3
4
-0,08
-0,13
R / a.u.
5
6
7
Correlation potential of magnesium (Roos-ATZPU basis set)
1 step calculations
VWN
LYP
0,13
OEP-MBPT(2)SD-f
OEP-MBPT(2)-f-1 shot
Correlation potential / a.u.
0,08
OEP-MBPT(2)-f-1 shot - HF
0,03
-0,02 0
0,5
1
1,5
2
2,5
-0,07
-0,12
-0,17
r / a.u.
3
3,5
4
4,5
Summary
Starting from a general theoretical framework based on the
density condition in Kohn-Sham theory and coupled cluster
theory, we have defined a rigorious exchange-correlation
functionals, potentials and orbitals.
We have performed an ab initio correlated dft calculations
employing the OEP-MBPT(2)
exchange and correlation
potentials.
We show the interconnections between the CC & MBPT
approach and DFT.
The calculations are fully self-consistent. (Except approximated
“one step” calculations)
The OEP-MBPT(2) correlation potentials and the exact correlation
potentials are in the excellent agreement with each other, while the
standard approximate DFT correlation potentials have a qualitatively
wrong behavior.
The total energies, and highest occupied orbital energies calculated by
OEP-MBPT(2) method are very accurate (CCSD(T) accuracy).
With OEP-MBPT(2) method we can treat with almost CCSD(T)
accuracy “week interactions” systems.
Our ab initio dft correlation potentials will be instrumental in
developing accurate and systematically improvable exchangecorrelation functionals and potentials.
The non expensive “one step” procedure which do not need self
consistent process can be very useful in obtaining approximate
correlation potentials
36
Some negative aspects
Exc and Vxc are orbital dependent
Strong basis set dependency of the LCAO-OEP results
Slow convergence in some cases
Numerical cost scales as Nit nocc2 nvirt3
Exchange and correlation potentials are very complicated –
they reflects the shell structure, changes in number of
particles
Ab initio dft (OEP) papers
•
•
•
•
•
•
•
•
S. Hirata, , S. Ivanov, I. Grabowski, R.J. Bartlett, K. Burke and J. Talman Is an OEP potential determined uniquely? J. Chem. Phys. 115 ,1635 ,(2001)
I. Grabowski, S. Hirata, S. Ivanov, R. J. Bartlett Ab‐initio density functional theory: OEP‐MBPT(2) – a new orbital‐dependent correlation functional.,
J. Chem. Phys. 116, 4415 (2002)
S. Hirata, S. Ivanov, I. Grabowski, R. J. Bartlett Time‐dependent density functional theory employing optimized effective potentials, J. Chem. Phys. 116, 6468, (2002)
S. Ivanov, S. Hirata, I. Grabowski, R. J. Bartlett Connections between Second‐
Order Gorling‐Levy and Many Body perturbation Approaches in Density Functional Theory. J. Chem. Phys. 118, 461 (2003)
R. J. Bartlett , I. Grabowski, S. Hirata, S. Ivanov, The Exchange‐Correlation Potential in ab initio Density Functional Theory. J. Chem. Phys. 122, 034104 (2005)
V. Lotrich, I.Grabowski, R.J. Bartlett Intermolecular potential energy surfaces of
weakly bound dimers computed from ab initio dft: the right answer for the right
reason. Chem. Phys. Lett. xxx, (2004)
I. Grabowski, V. Lotrich Acurate orbital‐dependent correlation and exchange‐
correlation potentials from noniterative ab initio dft calculations. Mol. Phys. xxx (2005)
S. Hirata, S. Ivanov, R. J. Bartlett , I. Grabowski Exact‐Exchange time dependent density functional theory for frequency‐dependent polarizabilities. ,
Phys. Rev. A 71, 1, (2005) 38