Density Functional Resonance Theory of Unbound Electronic Systems
DDensity Functional Resonance Theory of Unbound Electronic Systems
Daniel L. Whitenack ∗ Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA
Adam Wasserman † Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, IN 47907, USA andDepartment of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA (Dated: October 24, 2018)Density Functional Resonance Theory (DFRT) is a complex-scaled version of ground-state Den-sity Functional Theory (DFT) that allows one to calculate the resonance energies and lifetimes ofmetastable anions. In this formalism, the exact energy and lifetime of the lowest-energy resonance ofunbound systems is encoded into a complex “density” that can be obtained via complex-coordinatescaling. This complex density is used as the primary variable in a DFRT calculation just as theground-state density would be used as the primary variable in DFT. As in DFT, there exists amapping of the N -electron interacting system to a Kohn-Sham system of N non-interacting par-ticles in DFRT. This mapping facilitates self consistent calculations with an initial guess for thecomplex density, as illustrated with an exactly-solvable model system. Whereas DFRT yields inprinciple the exact resonance energy and lifetime of the interacting system, we find that neglect-ing the complex-correlation contribution leads to errors of similar magnitude to those of standardscattering close-coupling calculations under the bound-state approximation. Density Functional Theory (DFT) [1–3] provides oneof the most accurate and reliable methods to calculatethe ground-state electronic properties of molecules, clus-ters, and materials from first principles. It is one ofthe workhorses of computational quantum chemistry [4].In addition, DFT’s time-dependent extension (TDDFT)[5] can now be applied to a wealth of excited-state andtime-dependent properties in both linear and non-linearregimes [6]. When the N -electron system of interesthas no bound ground state, however, neither DFT norTDDFT can be applied in a straightforward way. A cor-rect DFT calculation converges to the true ground stateby ionizing the system, thus leaving no reliable startingpoint for a subsequent TDDFT calculation on the N -electron system. In practice, a finite simulation box orbasis set can make the system artificially bound [7, 8],but information about the relevant lifetimes is lost in theprocess.We address here this fundamental limitation of ground-state DFT, and propose a solution.Consider a system of N interacting electrons in anexternal potential ˜ v ( r ), with ground-state density ˜ n ( r ).The potential is set to be everywhere positive and go toa positive constant C as | r | → ∞ . The ground-state en-ergy is ˜ E >
0. We start by asking how the gound statedensity changes when a smooth step is added to ˜ v ( r ) ata radius | R | that is larger than the range of ˜ v ( r ). Thestep is such that the new potential v ( r ) coincides with˜ v ( r ) for | r | < | R | but goes to zero at infinity. Since ˜ v ( r )is everywhere positive, all N electrons tunnel out and v ( r ) supports no bound states. The correct ground stateenergy is now E = 0, and the new density n ( r ) is delocal-ized through all space. In practical calculations, however, v ( r ) and ˜ v ( r ) cannot be distinguished if | R | is beyondthe size of the simulation box. The result provided by ground-state DFT using the exact exchange-correlationfunctional is not E , but ˜ E >
0, and the density obtainedis ˜ n ( r ) as if the system were bound. Even when the sim-ulation box is large enough to include the steps, use of afinite basis-set of localized functions will artificially bindall electrons. Clearly, such calculations do not provideapproximations to the true ground-state energy and den-sity of v ( r ), but to those of its lowest-energy resonance(LER).The purpose of this letter is to establish an analogof KS-DFT that provides the in-principle exact LER-density along with its energy and lifetime for any finite | R | . As | R | → ∞ , the results coincide with those of stan-dard KS-DFT. For higher-energy resonances, TDDFT isneeded as a matter of principle [9, 10].First, we note that as | R | → ∞ , the complex density n θ ( r ) associated with the LER ofˆ H v = ˆ T + ˆ V ee + (cid:90) d r ˆ n ( r ) v ( r ) , (1)becomes equal to the complex density ˜ n θ ( r ) associatedto ˜ v ( r e iθ ). In Eq. 1, ˆ T = − (cid:80) Ni =1 ∇ i is the kineticenergy operator, ˆ V ee = (cid:80) Ni,j (cid:54) = i | r i − r j | − is the electron-electron interaction, and ˆ n ( r ) = (cid:80) Ni =1 δ ( r − ˆ r i ) is thedensity operator. (Atomic units are used throughout).To find n θ ( r ), we complex-scale ˆ H v by multiplying allelectron coordinates by the phase factor e iθ , diagonalizethe resulting non-hermitian operator ˆ H θv , and calculatethe bi-expectation value of ˆ n ( r ) as: n θ ( r ) = (cid:104) Ψ Lθ | ˆ n ( r ) | Ψ Rθ (cid:105) , (2)where | Ψ Rθ (cid:105) and (cid:104) Ψ Lθ | are the right and left eigenstatescorresponding to the complex eigenvalue of ˆ H θv that has a r X i v : . [ qu a n t - ph ] J un the smallest positive real part among all eigenvalues inthe non-rotating spectrum of ˆ H θv . For a detailed reviewof this technique and related methods in non-hermitianQuantum Mechanics, see ref. [11, 12]. The computationalcost of this prescription scales exponentially with thenumber of particles. Since n θ ( r ) → ˜ n θ ( r ) as | R | → ∞ ,and since there is a one-to-one correspondence between n θ ( r ) and v ( r e iθ ) [13, 14], the complex energy of the LER E θ [ n θ ] goes to ˜ E (not E ), as | R | → ∞ . Its lifetime L isgiven by ( − E θ )) − , and for any finite | R | , E θ [ n θ ] = E [ n θ ] − i L − [ n θ ] , (3)where the resonance energy E tends to ˜ E as | R | → ∞ .To build a complex analog of Kohn-Sham DFT us-ing n θ ( r ) as the basic variable, we first map the sys-tem of interacting electrons whose LER density is n θ ( r )to one of N particles moving independently in a com-plex “Kohn-Sham” potential v θs ( r ) defined such that its N occupied complex orbitals { φ θi ( r ) } yield the interact-ing LER-density via n θ ( r ) = (cid:80) Ni =1 (cid:104) φ θ,Li | ˆ n ( r ) | φ θ,Ri (cid:105) . Thecomplex Kohn-Sham equations are: (cid:18) ˆ h − ε i − ˆ h − τ − i ˆ h + 2 τ − i ˆ h − ε i (cid:19) (cid:18) Re( φ θi )Im( φ θi ) (cid:19) = 0 , (4)where ˆ h = − cos(2 θ ) ∇ + Re( v θs ( r )), and ˆ h = sin(2 θ ) ∇ + Im( v θs ( r )). The set of { ε i } and { τ i } pro-vide the orbital resonance energies and lifetimes of theKohn-Sham particles.Second, we write E θ [ n θ ] as: E θ [ n θ ] = T θs [ n θ ] + (cid:90) d r n θ ( r ) v ( r e iθ )+ E θ H [ n θ ] + E θ XC [ n θ ] (5)in analogy to standard KS-DFT, and require: T θs [ n θ ] = e − iθ T s [ n θ ] and E θ H [ n θ ] = e − iθ E H [ n θ ], where T s [ n θ ] and E H [ n θ ] are the standard non-interacting kinetic energyand Hartree functionals evaluated at the complex densi-ties. Eq. 5 then defines E θ XC [ n θ ]. The complex variationalprinciple [12] along with the assumption that the orbitalsused to construct the density can be expanded in an or-thonormal basis leads to the Euler-Lagrange equation: δE θ [ n θ ] δn θ − µ (cid:90) d r n θ ( r ) = 0 . (6)Performing the variarion in Eq. 5 and comparing withEq. 4 leads to an expression for the Kohn-Sham potentialthat is again analogous to that of standard KS-DFT: v θs ( r ) = v ( r e iθ ) + e − iθ v H [ n θ ]( r ) + v θ XC [ n θ ]( r ) , (7)where v θ XC [ n θ ]( r ) = δE θ XC [ n θ ] /δn θ ( r ) | LER .The simplest case where all essential aspects of this for-malism can be illustrated is a system of two interacting electrons moving in a one-dimensional potential such asthe one depicted in the inset of Fig. 1. We study a Hamil-tonian where the electrons interact via a soft-Coulombpotential of strength λ :ˆ H = (cid:88) i =1 (cid:20) − d dx i + v ( x i ) (cid:21) + λ (cid:112) x − x ) , (8)using v ( x ) = a (cid:34) (cid:80) j =1 (cid:16) e − c ( x +( − j d ) (cid:17) − − e − x b (cid:35) . Itsparent potential ˜ v ( x ) = a (1 − e − x /b ) goes to a as x →±∞ , but v ( x ) goes down to zero at x ∼ ± d . Exact solution via 2-electron wavefunction : Thecomplex-scaled Hamiltonian ˆ H θ = ˆ H ( { x i } → { x i e iθ } )was diagonalized with the Fourier Grid Hamiltonian(FGH) [15] and Finite Difference Methods. The numeri-cally exact n θ ( x ) was calculated via Eq. 2. The complexdensity n θ ( x ) depends on the value of θ (see Fig. 1),but for a large enough number of grid points the energydoes not. In the complex-scaling method the resonanceenergies are precisely those that remain stationary as θ changes [12]. Fig. 2 shows the energy for 0 < λ < Exact KS solution : Two non-interacting electrons inthe potential indicated by solid lines in Fig.3 have thesame n θ ( x ) as calculated above to one part in 10 (inthe sense that the space integral of the square of thedifference between their real or imaginary parts is lessthan 10 ). When n θ ( x ) is set to integrate to the numberof electrons (2, here), we verify this potential is given by: v θs ( x ) = e − iθ ∇ (cid:112) n θ ( x )2 (cid:112) n θ ( x ) − ε H + 2 iτ − H , (9)where ε H − iτ − H is the highest occupied complex orbitalenergy (in this case the only one), in exact analogy toreal KS-potentials for bound 2-electron systems. With-out an explicit expression for E θ XC [ n θ ], however, the totalenergy cannot be calculated via Eq. 5. Related work byErnzerhof [13] and physical intuition suggest that boundground-state functionals are applicable here. They are,in any case, the most natural candidates. Exchange : Borrowing knowledge from bound 2-electron DFT, Eqs. 4 and 7 were solved employing E θ X [ n θ ] = − E θ H [ n θ ] = − e − iθ E H [ n θ ]. The complex KSequations can be solved self-consistently with an initialguess for n θ . Using the non-interacting complex density,the SCF calculations converged in 4-5 iterations. The re-sulting complex energies are plotted in Fig. 2 along withthe exact results. For comparison, we also plot the re-sults from perturbation theory to first order in λ . Thetwo yield identical answers for the resonance energies,and extremely close for the lifetimes for all λ in the range0 < λ <
1. Thus, neglecting correlation, we find the av-erage error is ∼
14% for the real part and ∼
35% for theimaginary part of the total energy. We also compare with
FIG. 1. Different exact 2-electron complex densities whenusing different scaling angles. The model potential, v ( x ), usedin this study is also shown ( a = 4, b = 0 . c = 4, and d = 2).Grid θ Re( E ) Im( E )(N = 299) 0.27 4 . − . . − . . − . . − . . − . . − . θ practi-cally disappears. ( a = 4, b = 0 . c = 4, d = 2 and λ = 1) standard scattering calculations using the close-couplingequations under the bound state approximation [16, 17].The resonance energy is predicted by this method withan error of 22%, comparable to our DFRT exchange-onlyresults.As in standard KS-DFT, total energies are given hereby: E θ [ n θ ] = N (cid:88) i =1 (cid:0) ε i − iτ − i (cid:1) + E θ HX [ n θ ] − (cid:90) d r v θ HX n θ ( r ) (10)We point out that the θ -independence of the energy ispreserved by the SCF procedure (see Table I). As thegrid-size increases the dependence on θ becomes negli-gible. This is important, because within a SCF DFRTcalculation one is always solving the 1-body complex KS-equations. For these equations, one should be able to ef-ficiently use a large enough basis set or a fine enough gridto extinguish most of the numerical θ dependence. Thus,this well-known drawback of the complex-scaling tech-nique [18–20] is outdone by the benefit of never havingto deal with N -particle wavefunctions, but just 1-body(complex) densities. Correlation Potential : It is of interest to calculate theexact correlation potential, which we do by subtracting λ R e ( E θ ) ExactSCF, λ (1) λ I m ( E θ ) λ (1) ExactSCF
FIG. 2. The real and imaginary parts of E θ in the modelHamiltonian of Eq. 8 calculated exactly with complex scaling(thick solid), a first order correction to the non-interactingenergy (dashed), and our DFRT exchange-only self-consistentmethod (thin solid). ( a = 4, b = 0 . c = 4, and d = 2) the hartree-exchange contribution from the exact KS po-tential. The individual Hartree-exchange and correlationpotentials are shown in Fig. 4. To interpret the featuresin these complex potentials it is useful to distinguishbetween two regions. As the interaction between elec-trons is turned on and λ increases from 0 to 1, the regionaround the central well is shifted up in the real part ofthe Kohn-Sham potential. This behavior is also seen instandard KS-DFT, and serves to shift up the positionof the non-interacting orbital energies (in that case thereal part of the orbital energies). However, both the realand imaginary part of the complex Kohn-Sham potentialhave a second region outside the central well that showsa dramatic oscillatory structure arising purely from thefact that the state is unbound. It is already known thatthe decaying oscillations in the tails of the complex LERwavefunction are governed by the lifetime of the reso-nance [21]. These oscillations serve to produce the correctassymptotic behavior in the interacting complex densityand thereby give the correct interacting lifetime whenthis density is used in the functional.The analog of Koopmans’ theorem does not hold inDFRT. Although the ionization energy of our 2-electronsystem is strictly zero, it is tempting to define I θ ≡ E θ ( N = 1) − E θ ( N = 2) and check whether it equalsthe highest occupied KS orbital energy. For the param-eters used in Figs.1-4, E θ ( N = 1) = 1 . − . i , E θ ( N = 2) = 4 . − . i , but the exact KS eigenvalueis 2 . − . i . Clearly, DFRT provides an unambigu-ous prescription for the calculation of negative electronaffinities.We are working on the implementation of DFRT to cal-culate the lifetime of molecular metastable anions. Themethod would also be applicable to molecules connectedto metallic leads, as in molecular electronics. Ernzer-hof and co-workers have developed an approach for thatpurpose where complex absorbing potentials are addedwithin a complex-DFT framework [13, 22]. However, weemphasize that the complex potentials in DFRT are theresult of a variational calculation, and they are obtainedself-consistently for the N -electron system treated as iso- −5 0 5024 x R e ( v s θ ) −5 0 5−10 x I m ( v s θ ) FIG. 3. The real and imaginary part of the complex Kohn-Sham potential for the LER of 2 soft-Coulomb interactingelectrons in the model potential, Eq.8. The dashed lines arethe real and imaginary part of the complex-scaled parent po-tential ˜ v ( x ). ( θ = 0 . a = 4, b = 0 . c = 4, d = 2, and λ = 1). −5 0 500.20.4 x R e ( v H X θ ) λ =1 λ =0.75 λ =0.5 λ =0.25 λ =0 −5 0 500.04 x I m ( v H X θ ) λ =1 λ =0.5 λ =0.25 λ =0 λ =0.75 −5 0 5−1−0.500.5 x R e ( v c θ ) λ =0 λ =1 λ =0.5 λ =0.25 λ =0.75 −5 0 5−1−0.50 x I m ( v c θ ) λ =0 λ =0.25 λ =0.5 λ =0.75 λ =1 FIG. 4. The individual contributions to the Kohn-Sham po-tential from Hartree-exchange and correlation. ( θ = 0 . a = 4, b = 0 . c = 4, and d = 2) lated, rather than added to the Hamiltonian from thestart to model an open system.In addition, DFRT should be applicable to study shapeand Feshbach resonances in low-energy electron scatter-ing processes [23–25] of growing interest in biologicalsystems [26–28], atmospheric sciences, lasers, and astro-physics [29–32].In summary, DFRT provides an unambiguous prescrip-tion for calculating negative electron affinities based ona complex-scaled version of standard ground state DFT.This complex-scaled version has been cast in a way thatis analogous in practice to KS-DFT. Results on a modelsystem suggest that the same machinery that has beendeveloped for KS-DFT yields accurate resonance energiesand lifetimes in DFRT. It remains to be seen if commonapproximations to E XC [ n ] are able to capture the impor-tant effects that determine properties of real transientanions. A more detailed study of the complex densityfunction and various DFRT identities is forthcoming.Acknowledgment is made to the Donors of the Ameri- can Chemical Society Petroleum Research Fund for sup-port of this research under grant No.PRF ∗ † [email protected][1] P. Hohenberg and W. Kohn, Phys. Rev., , B864(1964).[2] W. Kohn and L. J. Sham, Phys. Rev., , A1133 (1965).[3] R. G. Parr and W. Yang, Density-Functional Theory ofAtoms and Molecules (Oxford University Press, Oxford,1994).[4] R. M. Martin,
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