AB INITIO DENSITY FUNCTIONAL THEORY - University Of Florida

1y ago
9 Views
2 Downloads
528.28 KB
89 Pages
Last View : 2m ago
Last Download : 2m ago
Upload by : Gideon Hoey
Transcription

AB INITIO DENSITY FUNCTIONAL THEORYByIGOR VITALYEVICH SCHWEIGERTA DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYUNIVERSITY OF FLORIDA2005

Copyright 2005byIgor Vitalyevich Schweigert

TABLE OF CONTENTSpageLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viiCHAPTER1INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.11.21.31.41.523.39121415EXACT ORBITAL-DEPENDENT EXCHANGE FUNCTIONAL . . . .172.12.22.3Exact Exchange Functional . . . . . . . . . . . . . . . . . . . . . .Optimized Effective Potential Method . . . . . . . . . . . . . . . .Performance of the Auxiliary-Basis EXX Method . . . . . . . . .171924CORRELATION FUNCTIONALS FROM SECOND-ORDERPERTURBATION THEORY . . . . . . . . . . . . . . . . . . . . . . .313.13.2.31.333940OTHER THEORETICAL AND NUMERICAL RESULTS . . . . . . . .484.14.2. .48. . .4955. .59CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654.34.45.3.33.44Ab-Initio Wavefunction-Based Methods .Kohn-Sham Density Functional Theory .Problems with Conventional FunctionalsOrbital-Dependent Functionals . . . . . .Ab initio Density Functional Theory . .1Correlation Functional from Second-Order Perturbation TheoryCorrelation functional from Second-Order Perturbation Theorywith Partial Infinite-Order Resummation . . . . . . . . . . . .Implementation of the PT2 and PT2SC Functionals . . . . . . .Numerical Tests for Ab initio Functionals . . . . . . . . . . . . .Connection between Energy, Density, and Potential . . . . . .Diagrammatic Derivation of the Optimized Effective PotentialEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mixing Exact Nonlocal and Local Exchange . . . . . . . . . .Second-Order Potential within Common Energy DenominatorApproximation . . . . . . . . . . . . . . . . . . . . . . . . .iii

APPENDIXAFUNCTIONAL DERIVATIVE VIA THE CHAIN RULE . . . . . . . . .69BSINGULAR VALUE DECOMPOSITION . . . . . . . . . . . . . . . . .73CDERIVATIVE OF THE SECOND-ORDER CORRELATION ENERGIES 75REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . .81iv

TableLIST OF TABLESpage2–1 Effect of basis set on the performance of the EXX method. . . . . . .262–2 Effect of the explicit asymptotic term on the performance of the EXXmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272–3 Effect of the Singular Value Decomposition threshold on theperformance of the EXX method . . . . . . . . . . . . . . . . . . . .282–4 Performance of the EXX methods for the 35 closed-shell molecules ofthe G1 test set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303–1 Performance of ab initio and conventional correlation functionals inthe high-density limit . . . . . . . . . . . . . . . . . . . . . . . . . .423–2 Performance of ab initio correlation functionals for closed-shell atoms .423–3 Density moments of Ne calculated with ab initio DFT, ab initiowavefunction and conventional DFT methods . . . . . . . . . . . . .464–1 Performance of the hybrid ab initio functional EXX-PT2h withoptimized fraction of nonlocal exchange . . . . . . . . . . . . . . . .57v

FigureLIST OF FIGURESpage2–1 Explicit asymptotic terms for Ne and the corresponding EXXpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283–1 Performance of ab initio DFT and ab initio wavefunction methods intotal energy calculations for the G1 test set. . . . . . . . . . . . . .433–2 Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in calculations of the total energy asa function of the bond lengths . . . . . . . . . . . . . . . . . . . . .453–3 Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in dipole moment calculations for theG1 test set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474–1 The total energy and first density moment of Be calculated withthe EXX-PT2h functional with various fractions of the nonlocalexchange operator . . . . . . . . . . . . . . . . . . . . . . . . . . .58vi

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of PhilosophyAB INITIO DENSITY FUNCTIONAL THEORYByIgor Vitalyevich SchweigertAugust 2005Chair: Rodney J. BartlettMajor Department: ChemistryAb initio Density Functional Theory (DFT) is a new approach to theelectronic structure problem that combines elements of both density-functionaland wavefunction-based approaches. It avoids the limitations of conventionalDFT methods by using orbital-dependent functionals based on the systematicapproximations of wavefunction theory.The starting point of ab initio DFT is the exact exchange functional. Thisfunctional was implemented with the auxiliary-basis Optimized Effective Potentialmethod. The effect of numerical parameters on the performance of the method wasalso examined.It has been suggested in the literature to use perturbation theory to constructthe correlation counterpart of the exact exchange functional. In this study,an ab initio correlation functional from second-order perturbation theory wasimplemented. However, numerical tests showed that this functional fails to providean adequate description of correlation effects in molecules. This problem wasattributed to the poor convergence of the perturbation series based on the KohnSham determinant and a partial infinite-order resummation of one-body terms wasvii

proposed as a solution. The new functional offers a more balanced description ofcorrelation effects, as was demonstrated in applications to a number of closed-shellatoms and molecules. It resulted in energies and densities superior to conventional(Møller-Plesset) second-order perturbation theory or DFT methods, accuratelyreproduced potential energy surfaces, and led to qualitatively correct effectivepotentials and single-electron spectra.An extension of the method based on mixing exact local and nonlocalexchange and an approximate second-order correlation potential were alsoexamined.viii

CHAPTER 1INTRODUCTIONThe underlying physical laws necessary for the mathematical theory ofa large part of physics and the whole of chemistry are thus completelyknown, and the difficulty is only that the exact application of these lawsleads to equations much too complicated to be soluble.P. A. M. Dirac, Proc. Roy. Soc. London, p. 174, 1929At the microscopic level, a chemical reaction is the transition from one stableconglomerate of nuclei and electrons (reagent) to another one (product). Giventhe initial configuration of the system, the transition properties and the finalstate are determined by the interactions of the particles with each other and withthe environment. Since the nature of these interaction is known, it is then thetask of theoretical chemistry to predict the outcome of the reaction by solvingthe fundamental equation describing these particles. Dirac’s famous words statethe ultimate goal of theoretical chemistry — the complete substitution of theexperiment by a theoretical calculation — and warn about the ultimate difficulty— the immense complexity of the problem. Even now, given all the computationalpower at our disposal, the near-exact solutions of the electronic problem are stilllimited to few-electron systems.Facing the intractability of the exact solution, one must rely onapproximations. Although, for systems beyond several thousands particles onehas no choice but to rely on classical mechanics, most chemical phenomena requirea quantum-mechanical description to obtain at least qualitative resemblancewith reality. In quantum theory, a chemical system is described by the molecular1

2Hamiltonian (neglecting magnetic and relativistic effects for simplicity)Ĥ elec.X1i 12 2i nucl.XA 1elec. nucl.elec.nucl.XX1 2 XXZA1Z Z A B . A 2r i RAri rjRA RB ii jAA B(1-1)The solution of the Shrödinger equation information about the system evolution.ĤΨ(t; x1 , ., xN ) i Ψ(t; x1 , ., xN ) t(1-2)Thus, quantum chemistry, the collection of quantum-mechanical methods to solveEq. 1-2, plays the major role in theoretical chemistry.Ab initio Density Functional Theory belongs to the class of quantum-chemicalmethods called electronic structure methods. These methods further simplifythe Hamiltonian of Eq. 1-1 using the Born-Oppenheimer approximation. In thisapproximation one neglects the coupling between the electronic and nucleardegrees of freedom, which allows one to factorize the corresponding variables andconcentrate on the electronic part of the wavefunction. Also, if one is interestedin stationary solutions, the time variable can also be factorized. Thus, the timeindependent electronic structure problem is to find the solution of the timeindependent Schrödinger equationĤΨ(x1 , ., xN ) EΨ(x1 , ., xN )(1-3)defined by the nonrelativistic, Born-Oppenheimer electronic HamiltonianĤ elec.X1i2 2i elec. nucl.XXiAelec.XZA1 r i RA r i r j i j(1-4)Ab initio DFT is a new method for obtaining approximate solutions to Eq. 1-3.Its formalism is based on two fundamental electronic structure approaches : abinitio wavefunction methods and Density Functional Theory.

31.1 Ab-Initio Wavefunction-Based MethodsHartree-Fock Method.The simplest approximate wavefunction that retains the correct fermionsymmetry is given by the antisymmetric product of single-electron wavefunctions Φ(x1 , ., xN ) (N !)1/2  φ1 (x1 ).φN (xN ) ,where X( 1)σP P̂(1-5)(1-6)Pensures that Φ is antisymmetric with respect to a permutation of the labels ofany pair of electrons. This type of wavefunction can be conveniently written as adeterminantΦHF φ1 (r 1 ) . . . φ1 (r N ) . . . φN (r 1 ) . . . φN (r N ) (1-7)and is often called a Slater determinant or single-determinant wavefunction.In the Hartree-Fock method, the single-electron wavefunctions (or orbitals)are determined by the condition that the corresponding determinant minimizes theexpectation value of the true many-electron Hamiltonian [1] EHF ΦHF Ĥ ΦHF min Φ̃ Ĥ Φ̃ ,(1-8)Φ̃ subject to the constraint that the orbitals remain orthonormal, φp φq δpq .Inserting the expression for Φ from Eq. 1-5 into this expectation value, oneobtains the expression for the Hartree-Fock energy in terms of the orbitalsEHFelec.elec. 1X X 1 2 φi vext φp ΦHF Ĥ ΦHF ij ij22iij (1-9)

4 where ij ij is the Dirac notation for the two-electron integrals defined byEq. 1-10. ij ij ij ij ij jiZZφ (r)φ j (r 0 )φi (r)φj (r 0 )φ (r)φ j (r 0 )φj (r)φi (r 0 )0 i0 i drdr drr r r 0 r r 0 (1-10)Requiring that EHF be stationary with respect to an arbitrary variation of{φi } one obtains the Hartree-Fock equations X 1( 2 vext vH v̂nlx ) φp ²pq φq2qwherevext (r) nucl.XAZA r R A (1-11)(1-12)is the external Coulomb field created by the nuclei andZvH (r) ρ(r 0 ) dr 0 r r0 (1-13)is the Hartree potential (i.e., the Coulomb field created by the total electrondensity),ρ(r) elec.Xφ i (r)φi (r),(1-14)iv̂nlx is the nonlocal exchange operator,elec. X r v̂nlx φp φi (r)iZdr 0φ i (r 0 )φp (r 0 ) r r 0 ,(1-15)and ²pq are the Lagrange multipliers that ensure the orthonormality of the HartreeFock orbitals.Note that the number of solutions of Eq. 1-11 is not limited to the number ofelectrons. The lowest N solutions are referred to as occupied orbitals (N being thenumber of electrons) and the remaining solutions are referred to as virtual orbitals.

5Using the fact that the Fock operator1fˆ 2 vext vH v̂nlx2(1-16)is invariant with respect to any unitary transformation of the occupied orbitals, onecan transform Eq. 1-11 to its canonical form, fˆ φp ²p φp ,(1-17)which is the eigenvalue problem for the Fock operator.Since the Fock operator depends on {φi } through the vH and v̂nlx , Eq. 1-11 isan integro-differential equation that can be solved iteratively, until self-consistencyis reached. Therefore, the Hartree-Fock approximation belongs to the class ofSelf-Consistent Field (SCF) approximations.One can solve the Hartree-Fock equations numerically. However, a morepractical approach is to use a finite basis set (usually atom-centered Gaussian-typefunctions) to expand the HF orbitals. As the result, the Hartree-Fock integrodifferential equations are transformed into a matrix problem.Electron-Correlation Methods.The Hartree-Fock method can recover as much as 99% of the total electronicenergy. Still, even the remaining error of 1% is too large on the chemical scale andmay lead to a qualitatively wrong theoretical prediction.The difference between the SCF and exact solutions is due to electroncorrelation effects. In ab initio electron-correlation methods, one relies on elaboratemany-body techniques to go beyond the SCF approximation and account for thesimultaneous electron-electron interactions. These methods, in contrast to therelatively simple Hartree-Fock approximation, can be quite challenging conceptuallyand computationally.

6The correlation limit (i.e., the exact solution of Eq. 1-3 in a given basis set)can be obtained via the Full Configuration Interaction method. In this method, thecorrelation correction to the Hartree-Fock determinant is expanded over all possibleexcited determinantsΨF CI ΦHF occ. Xvirt.XiaCia Φai occ. Xvirt.XCij C ab Φabij . . .(1-18)i6 j a6 bwhere Φai , Φabij , etc. are formed by by substituting several occupied orbitals in theHartree-Fock determinant by virtual orbitals, e.g. Φai (N !)1/2 Â φ1 (x1 ).φa (xi ).φN (xN ) .(1-19)The expansion coefficients are found from the variational condition on theexpectation value of the true Hamiltonian ΨF CI Ĥ ΨF CI EF CI minab ,.ΨF CI ΨF CICia ,Cij(1-20)However, the number of possible excited determinants grows exponentiallywith the number of electrons and basis functions, therefore, the Full CI methodis computationally intractable for any but very small systems. Among theapproximate electron-correlation methods, the most common are the truncatedand multi-reference Configuration Interaction methods, Coupled-Cluster methods[2], and Many-Body Perturbation Theory [3]. For example, for systems wherethe multi-reference treatment is not necessary (i.e., when the Hartree-Fockwavefunction dominates the Full CI expansion), the Coupled-Cluster methodshave proved to be the most systematic and computationally robust approach to themany-electron problem.

7Many-Body Perturbation Theory.In some cases, perturbation theory can provide an accurate description ofelectron-correlation effects at a significantly lower cost than required by CoupledCluster or multireference methods. For example, second-order Rayleigh-Schrödingerperturbation theory is the simplest and least expensive ab initio method forelectron correlation. That is why it was chosen as the basis for the ab initiocorrelation functional (Chapter 3).In such perturbation theory, one finds the solution of the many-body problem(Eq.1-21) using an SCF model (Equations 1-22 and 1-23) as the reference. Ĥ Ψ E Ψ(1-21) 1( 2 û) φp ²p φp .2(1-22)elec. X 1 H0 Φ ( 2 û Φ E0 Φ,2i(1-23)where Φ is the single-determinant wavefunction constructed from the N lowestsolutions to Eq. 1-22. The remaining eigenfunctions of Ĥ0 are obtained bysubstituting the corresponding number of occupied orbitals in Φ by the virtualorbitals.To do this, the true Hamiltonian is partitioned into the reference Hamiltonianand perturbationĤ H0 V̂whereV̂ Ĥ Ĥ0 elec.Xivext (r i ) elecXi6 j(1-24)elec.X1 û r i r j iThe solution to Eq. 1-21 is then found by introducing the perturbationparameter λ and expressing the corrections to the reference wavefunction and(1-25)

8energy as series of terms of increasing powers of λĤ Ĥ0 λV(1-26) Ψ Φ λ Ψ(1) λ2 Ψ(2) . . .(1-27)E E0 λE (1) λ2 E (2) . . .(1-28)These order-by-order corrections can be found by neglecting all higher termsfrom the Schrödinger equation (E0 H0 ) Ψ(1) (V̂ E (1) ) Φ(1-29) (E0 H0 ) Ψ(2) (V̂ E (1) ) Ψ(1) E (2) Φ(1-30)and so forth.Choosing the perturbative corrections to be orthogonal to the reference wavefunction, Ψ(n) Φ 0, one can readily obtain the expressions for the orderby-order contributions to the energy by projecting the Equations 1-29 and 1-30onto the reference space E (1) Φ V Φ(1-31) E (2) Φ V Φ(1)(1-32)The order-by-order contributions to the wavefunction can be written in termsof the resolvent operator [4] (the inverse of integro-differential operator E0 H0 inthe Hilbert subspace) (1) Ψ R̂0 V Φ(1-33)

9 (2) Ψ R̂0 (V̂ E1 ) Ψ(1) R̂0 (V̂ E1 )R̂0 Φ ,(1-34)whereR̂0 Q̂,E0 H0(1-35) and Q̂ 1̂ Φ Φ is the projector onto the complementary space of Φ (Hilbert space with Φ excluded.)The actual expression for the resolvent operator can readily be found byrecognizing that E0 H0 is diagonal in terms of eigenfunctions of H0R̂0 all XΦn Φn n6 0E0 En(1-36)Note that the Hartree-Fock SCF model presents a special case as the referencefor the perturbation expansion. First, the HF energy is correct through first order E0 E (1) Φ H0 Φ Φ V Φ Φ H Φ EHF(1-37)Second, the HF SCF Hamiltonian cancels all the effective one-body terms of thetrue Hamiltonian, leaving only two-body terms in the perturbation, so that onlydouble-excited determinants contribute to the second-order energy 2occ. Xvirt. ij ab X(2)EHF ² ²j ²a ²bi,j a,b i(1-38)1.2 Kohn-Sham Density Functional TheoryDensity Functional Theory is an alternative approach to the electronicstructure problem of Eq. 1-3 that uses the electronic density rather than thewavefunction as the basic variable. The formal basis of DFT is provided by twotheorems introduced by Hohenberg and Kohn [5]. The first theorem establishesthe one-to-one correspondence between the electronic ground-state density andthe external potential. Since it is the external potential that defines a particular

10molecule, the existence of such a correspondence ensures that the ground-stateelectronic density alone carries all the information about the system. In particular,the ground-state energy can be written as a functional of the density. However,there is no equation of motion for the electronic density. Instead, one must rely onthe second Hohenberg-Kohn theorem that states that the ground-state energy as afunctional of the density is minimized by the true ground-state density. Therefore,given the energy functional, one can obtain the ground-state density and energy byvariational minimization of the functional.However, the formal definition of Density Functional Theory does not tellhow to construct such functional. Several approximate forms have been suggested;however, they are far from accurate. The kinetic energy of electrons is particularlydifficult to approximate as a functional of the density.The idea of Kohn and Sham [6] was to use a SCF model (Eq. 1-39) totransform the variational search over the density into a search over the SCForbitals that integrate to a given trial density.1[ 2 vs (r)]φp (r) ²p (r)2(1-39)Such a transformation does not restrict the variational space, provided that everyphysically meaningful density correspond to a unique set of SCF orbitals (thev-representability condition). Not only does the use of the Kohn-Sham SCF modelensure that the variational search be to fermionic densities, but also it providesa good approximation for the kinetic energy. Indeed, provided that the orbitalsintegrate to the exact density, the so-called noninteracting kinetic energyocc.elec. X1 2 X 1 2 φi φi Φs Ts Φs 22iishould account for a large part of the true kinetic energy.(1-40)

11The remaining unknown terms of the energy functional are grouped into theexchange-correlation functionalExc [ρ] E[ρ] Ts Eext EH(1-41)where EH is the Hartree energy, which (as well as Ts ) can be readily calculatedfor a given set of SCF orbitals. Since Ts should reproduce a large part of T ,this procedure eliminates the necessity to model the entire kinetic energy as afunctional of the density. Thus, it is expected that Exc is easier to approximate as afunctional of the density than E.Note that according to the definition of the exchange interaction inwavefunction theory, the exchange component of the exchange-correlationfunctional is defined as Ex Φs Vee Φs EH(1-42)and the correlation component is the remaining partEc Exc Ex(1-43)The Kohn-Sham SCF orbitals are defined by the effective potential vs .Transforming the variational condition on the energy functional into the conditionfor the constrained search over the orbitals, one can obtain δ E[ρ] Tsδ Eext EH Excvs (r) vext vH (r) vxc (r)δρ(r)δρ(r)(1-44)where the exchange-correlation potential is defined as the functional derivative ofthe exchange-correlation functionalvxc (r) δExcδρ(r)(1-45)Thus, given an exchange-correlation functional, one defines the exchangecorrelation potential and then solves the Kohn-Sham SCF equations. Note that

12since it is an SCF model, practical implementation of the KS procedure is verysimilar to the Hartree-Fock method. Usually, the SCF orbitals are expanded ina Gaussian-type atom-specific basis, which transforms the Kohn-Sham integrodifferential equation into a matrix SCF equation. After self-consistency is reached,the SCF orbitals are guaranteed to reproduce the true density of the many-electronsystem. Also, the true energy can be found by inserting this density into the energyfunctional.Virtually all modern implementations of DFT use the Kohn-Sham scheme.However, the theory still leaves open the question of how to construct theexchange-correlation functional. Therefore, the principal challenge for thetheoretical development of DFT remains the construction of accurate exchangecorrelation functionals.1.3 Problems with Conventional FunctionalsThe conventional approach is to approximate the energy functional as ananalytical expression of the density and its gradients. The effective potential canthen be obtained in analytical form as well, and the KS equations can be solvedreadily. This approach started with the simplest Local Density Approximation(LDA) where the energy is given through an integral of a local functionalof the density. The next-level, Generalized Gradient Approximation (GGA)functionals, improved on the LDA functional form by including the dependenceon the gradients of the density. This extension provided a certain freedom indefining the form of the functional, and a number of different forms have beensuggested. Typically, the basic form of a GGA functional is chosen to satisfya set of conditions known to be satisfied by the exact functional. The basicform is then either parameterized to reproduce experimental data (empiricalfunctionals) or further modified to satisfy an extended set of conditions (nonempirical functionals).

13With the conventional functionals, KS DFT surpasses the quality of the HFmethod, and becomes comparable with the simplest ab initio correlation methods.Nevertheless, restricting the functional form to analytical expressions of the densityimposes certain limitations on the energy functional. GGA exchange functionalsare not capable of complete elimination of the spurious self-interaction componentof the Hartree energy. Since the exchange part often dominates the exchangecorrelation energy, the self-interaction error can considerably reduce the accuracyof the GGA functional. Similarly, semilocal correlation functionals cannot describepure nonlocal components of the correlation energy such as dispersion. Thisomission greatly reduces the applicability of the conventional KS DFT methods toweakly-interacting systems.Another problem is that while the GGA functionals result in relativelyaccurate energies, the functional derivatives (i.e., the corresponding KS potentials)are not nearly as accurate, especially in the inter-shell and asymptotic regions.Consequently, one should not expect the same level of accuracy for the densityas for the energy. Furthermore, the qualitatively incorrect potentials reducesubstantially the usefulness of the KS orbitals and orbital energies, which are oftenused to calculate certain ground-state properties or as the basis for response andtime-dependent KS DFT calculations.Some of these problems can be addressed without extending the functionalform. Several post-SCF corrections have been suggested to partially remove theself-interaction error. For example, after the KS equations have been solved, onecan introduce corrections to the energy to include dispersion or ensure the correctasymptotic behavior of the KS potential. However, these corrections are specific tothe particular functional and class of systems and they likely are incompatible witheach other. Clearly, one needs to go beyond the GGA functional form to resolvethese problems in a consistent and universal fashion.

141.4 Orbital-Dependent FunctionalsIt has now been fully recognized that KS orbitals can provide extrainformation about the system that cannot be “extracted” easily from the density orits gradients. The next-generation functionals (hybrid- and meta-GGA) augmentthe GGA functional form with terms that depend explicitly on the orbital ratherthan the density.An alternative approach is to dismiss completely the conventional hierarchy ofapproximations and construct the functional using solely the orbitals. In contrastto the conventional functionals, orbital-dependent functionals are analyticalexpressions of the orbitals (and orbital eigenvalues). They still are implicitfunctionals of the density, however. Indeed, the central assumption of KS DFTis that there exists one-to-one mapping between the exact density and some a localpotential. Therefore, a given density uniquely defines the potential, which, in turn,uniquely defines the orbitals through the KS SCF equations. Therefore, the orbitalsand explicitly orbital-dependent functional are implicit functionals of the density.One can think of the KS orbitals as the intermediate step in the mapping from thedensity to the energy.The most significant difference between the orbital-dependent and conventionalfunctionals is how the corresponding potential (i.e., the functional derivative withrespect to the density) is determined. The conventional functionals are givenas analytical expressions in terms of the density. Therefore, one can take thefunctional derivative straightforwardly to obtain an analytical expression for thepotential. The orbital-dependent functionals are analytical expressions in termsof the orbitals, whose dependence on the density is given through the effectivepotential and KS integro-differential equation. Therefore, the analytical expressionfor the functional derivative (hence, potential) cannot be obtained directly. Instead,one must rely on the chain rule to obtain an integral equation for the potential

15(Chapter 2). This integral equation is identical to the one used in the OptimizedEffective Potential (OEP) method. The OEP method is, therefore, the cornerstoneof DFT with orbital-dependent functionals.The immediate advantage of the orbital-based approach is that the exactexchange functional is known in term of orbitals. One can think of the EXXmethod as an extension to the idea of Kohn and Sham, where SCF orbitals areused to calculate both the larger part of the kinetic energy and a (presumablylarger) part of the exchange-correlation energy.1.5 Ab initio Density Functional TheoryWhile the EXX functional provides the exact description of the exchangeinteractions, it is just a first step towards the exact exchange-correlation functional.It is the effective description of electron correlation effects that makes KS DFTa powerful alternative to the ab initio wavefunction methods. Thus, one needs acorrelation functional that can be combined with the EXX functional.Conventional (GGA or higher-level) correlation functionals are developedin combination with the corresponding approximate exchange functionals andoften compensate the deficiencies of the latter. For example, the GGA correlationfunctionals usually result in correlation potentials that have the opposite sign tothe exact one. The terms correcting the approximate exchange are “hidden” in thecorrelation functionals and inseparable from the “true” correlation terms.Thus, it is not surprising that substituting the approximate exchange byits exact counterpart destroys the balance between the approximate exchangeand approximate correlation components and results in a functional inferior tothe exchange-only approximation. In other words, the conventional correlationfunctionals are not compatible with the EXX functional. Thus, the primarychallenge in the orbital-based approach to exchange-correlation functional is

16to develop an orbital-dependent correlation functional that can be combinedseamlessly with the exact exchange functional.Ab initio DFT solves the problem of constructing an orbital-dependentcorrelation functional by referring to ab initio wavefunction methods. The ideais simple: the goal of ab initio methods is to calculate the correction to the exactexchange approximation (i.e., correlation energy) in terms of the SCF orbitals.Thus, such an

3{2 Performance of ab initio correlation functionals for closed-shell atoms . 42 3{3 Density moments of Ne calculated with ab initio DFT, ab initio wavefunction and conventional DFT methods . . . . . . . . . . . . . 46 4{1 Performance of the hybrid ab initio functional EXX-PT2h with

Related Documents:

A molecular picture of hydrophilic and hydrophobic interactions from ab initio density functional theory calculations J. Chem. Phys. 119, 7617 (2003); 10.1063/1.1617974 Adsorption and desorption of S on and off Si(001) studied by ab initio density functional theory J. Appl. Phys. 84, 6070 (1998); 10.1063/1.368918 Adsorption of CO on Rh(100 .

Perpendicular lines are lines that intersect at 90 degree angles. For example, To show that lines are perpendicular, a small square should be placed where the two lines intersect to indicate a 90 angle is formed. Segment AB̅̅̅̅ to the right is perpendicular to segment MN̅̅̅̅̅. We write AB̅̅̅̅ MN̅̅̅̅̅ Example 1: List the .

2.2. ab initio Molecular Orbital Calculations All the ab initio MO calculations were made with a Dell personal computer using the Gaussian 94W program package. Based on the results of a previous paper [5], the used ab initio MO theory and basis set were restricted to the Hartree-Fock self-consistent field (HF) method the-

T3. Ab Initio Hartree-Fock Calculations The point of the empirical parameters in semiempirical calculations was to cut down on the number of electron-electron repulsion integrals that needed to be computed. In ab initio calculations one simply calculates them all. This inevitably means that ab initio calculations take much longer than semiempirical

density functional (KEDF) to accurately and efficiently simulate various covalently bonded molecules and materials within orbital-free (OF) density functional theory (DFT). By using a local, density-dependent scale function, the total density is decomposed into a hi

INDONESIAN AB INITIO ￉ STANDARD LEVEL ￉ PAPER 1 INDON￉SIEN AB INITIO ￉ NIVEAU MOYEN ￉ ￉PREUVE 1 INDONESIO AB INITIO ￉ NIVEL MEDIO ￉ PRUEBA 1 Monday 6 May 2002 (morning) Lundi 6 mai 2002 (matin) Lunes 6 de mayo de 2002 (ma￉ana) 1 h 30 m!IB DIP

English ab initio Standard level Paper 1 Anglais ab initio Niveau moyen Épreuve 1 . Livret de questions et réponses Instructions destinées aux candidats Écrivez votre numéro de session dans les cases ci-d

The Female Reader: Occupying a Space of her Own Women formed a large and increasing part of the new novel-reading . public. The traditional discrepancy between male and female literacy rates was narrowed, and finally eliminated by the end of the nineteenth ; century. The gap had always been the widest at the lowest end of the social scale. In Lyons at the end of the eighteenth century, day .