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 Expensive, but provides results which are quaranteed to converge to the solution of the Schrödinger equation as electron correlation and basis set is extended. MBPT(2) < CCD < CCSD < CCSD(T) < CCSDTQ < FCI DFT Its one particle structure make it possible to treat much larger systems than WFT. Almost all unknown informations are contained in an exchange correlation functional Exc and its associated potential Vxc. 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 Density condition path taking functional derivative General theoretical with respect to density of framework based on the orbital dependent energy density condition in Kohn functional (from ab initio Sham theory involving WFT) to get exchangecorrelation potential in KS many body perturbation coupled cluster method, theory. theory, and technically MBPT(2) level diagramatic manipulation. Problems with extending to higher orders
Basic formalism functional derivative path The spin densities ρ σ (r) and KS orbitals {φ pσ (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 E xc OEP [{φ pσ }] 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 V XC in each KS(OEP) iteration,
Formally we can represent V xc as Where X sσ -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.
E xc =E x +E c E x - HF exchange energy functional in terms of KS orbitals V x 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: OEP- MBPT(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 φ i (r) have to vanish. ρ(r)= ρ KS (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 H 0 = OEP MBPT(2) f H 0 = + ε p{ p p} OEP MBPT(2) sc p p f pp + { p p} H 0 = + + + f pp{ p p} + fij{ i j} + fab{ a b} p 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 twoelectron integrals we calculate V XC, Going back with V XC 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 Correlation potential / 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)-KS OEP-MBPT(2)-f
21
22
Correlation potential of neon (Roos-ATZPU basis set) Correlation potential / a.u. 0,32 VWN 0,27 LYP exact correlation potential 0,22 OEP-MBPT(2)SD and D 0,17 OEP-MBPT(2)-SD-f 0,12 0,07 0,02-0,03 0 1 2 3 4 5 6 7 radial charge density -0,08-0,13 R / a.u.
0 Exchange-correlation potential of neon (Roos-ATZPU) one step calculations 0 1 2 3 4 5 6-1 -2 Potential / a.u. -3-4 -5-6 -7-8 exchange-correlation OEP-MBPt(2)-f and OEP-MBPT(2) `exchange-only' OEP SVWN BLYP OEP-MBPT(2)-f-1shot-HF -9 R / a.u.
Correlation potentials of Be atom 0,10 0,00 0 1 2 3 4 5 6 7 Correlation potential / a.u. -0,10-0,20-0,30 OEP-MBPT(2)-f vc exact OEP-MBPT(2)-KS - (non converged) LYP -0,40 VWN -0,50 R / a.u.
Energy surface of He2 (17s10p2d) 0,002 0,001 E-E / ev -0,001-0,002 3 4 5 6 7 8 9 10 11 12 MBPT(2) CCSDT -0,003 OEP-MBPT(2) PBE -0,004 r / au
Ne 2 dimer potential energy - AUG-CC-PVTZ basis set 300 Energy[cm -1 ] 250 200 150 100 50 0-50 MP2 CCSD CCSD(T) OEP_MBPT(2)-fsc 5 5,5 6 6,5 7 7,5 8-100 -150-200 r [a.u]
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 0,02 Correlation potential / a.u. 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 of neon 1 step calculations Correlation potential / a.u. 0,32 LYP 0,27 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 5 6 7 VWN -0,08-0,13 R / a.u.
Correlation potential of magnesium (Roos-ATZPU basis set) 1 step calculations VWN Correlation potential / a.u. 0,13 0,08 0,03-0,02-0,07 LYP OEP-MBPT(2)SD-f OEP-MBPT(2)-f-1 shot OEP-MBPT(2)-f-1 shot - HF 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5-0,12-0,17 r / a.u.
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 E xc and V xc are orbital dependent Strong basis set dependency of the LCAO-OEP results Slow convergence in some cases Numerical cost scales as N it n occ2 n 3 virt 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 exchangecorrelation 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