Letter pubs.acs.org/JPCL
Calculating the Lifetimes of Metastable States with Complex Density Functional Theory Yongxi Zhou and Matthias Ernzerhof* Département de Chimie, Université de Montréal, C.P. 6128 Succursale A, Montréal, Québec H3C 3J7, Canada ABSTRACT: Among other applications, complex absorbing potentials (CAPs) have proven to be useful tools in the theory of metastable states. They facilitate the conversion of unbound states of a finite lifetime into normalized bound states with a complex energy. Adding CAPs to a conventional Hamiltonian turns it into a nonHermitian operator. Recently, we introduced a complex density functional theory (CODFT) that extends the Kohn−Sham method to the realm of non-Hermitian systems. Here, we combine CAPs with CODFT and present the first application of CODFT to metastable systems. In particular, we consider the negative ions of the beryllium atom and the nitrogen molecule. Using conventional exchange−correlation functionals as functionals of a complex density, the resonance positions and the resonance lifetimes are obtained, and they are in line with the findings of other studies. SECTION: Molecular Structure, Quantum Chemistry, and General Theory
R
complex external potentials and complex densities. Within this formalism, with slight but crucial modifications, the conventional equations of KS DFT remain valid; yet, the electron density is replaced by its complex-valued generalization, and the external local potentials are complex, containing a real as well as an imaginary part. Recently, we implemented23 CODFT and applied it to electron transport. In this Letter, we present the first application of CODFT to metastable states of atoms and molecules by combining it with CAPs. CODFT has not been used before to calculate complex energies; it has therefore not yet been put to the test. After a short overview of CODFT, we demonstrate that meaningful results are obtained from this theory. It is worth stressing that conventional approximations to the exchange−correlation energy are employed as functionals of a generalized complex-valued density. Wasserman, Moiseyev, and Whitenack as well as Giraud and co-workers24−27 also provided extensions of the Hohenberg−Kohn theorems to unbound systems. These authors showed that resonances can be calculated by complex scaling combined with an appropriate extension of the ground-state DFT. Tozer and co-workers (see, e.g., refs 28 and 29) provided schemes for the calculation of the resonance position of metastable ions within conventional KS DFT. The addition of a CAP of the form iηW to a molecular Coulomb Hamiltonian H M results in a non-Hermitian Hamiltonian
esonant, unbound electronic states of molecules, clusters, and extended systems appear in various areas; they are, for instance, a focal point in electron−molecule scattering.1−8 Furthermore, resonances, which we also refer to as metastable states, are useful for the description of transient phenomena in molecular electronic devices. Several theories have been developed in the past to describe metastable electronic states. The work of Balslev, Aguilar, Combes, Balslev, and Simon9−11 produced the widely used complex coordinate scaling approach. Reviews of this formalism are, for instance, provided in refs 12−15. An alternative to the complex coordinate scaling are complex absorbing potentials (CAPs),16−19 which are added to the exterior region of the metastable system to absorb the wave function and make it square-integrable. It is then possible to calculate a normalized stationary state whose eigenvalue E = ERe − iΓ/2 has a real part, yielding the resonance position, and an imaginary part, which is inversely proportional to the lifetime of the system. Correlated electronic structure methods, such as configuration interaction (CI),20 Fock-space multireference coupled cluster (FSMRCC),21 and the Green’s function formalism,17 have been combined with the CAPs approach and applied to various small molecules. However, while the listed methods can be readily adapted to non-Hermitian Hamiltonians, they are computationally costly for large molecules. An efficient electronic structure method is provided by density functional theory (DFT). Yet, in DFT, it is not obvious how to deal with non-Hermitian operators; more precisely, the Hohenberg−Kohn theorems do not apply to nonHermitian operators. Nevertheless, recently, generalizations of DFT have been presented22−27 that are suitable for the direct calculation of complex eigenvalues. Here, we focus on complex density functional theory (CODFT),22,23 which is an extension of conventional Kohn−Sham (KS) DFT to the realm of © 2012 American Chemical Society
H(η) = HM − iηW
(1)
Received: May 25, 2012 Accepted: July 4, 2012 Published: July 10, 2012 1916
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The positive factor η adjusts the strength of the CAP, and W is a positive semidefinite, local, one-electron operator. Often, W is zero inside of a cubic box or spherical shell17 surrounding the target molecule, and it is nonzero only outside of this container. If a complete basis set could be used, the exact complex eigenvalue would be obtained by taking the limit η → 0+. In actual computations with finite basis sets, the resonance can be identified through a stability condition,16 yielding the optimum value of η
η
∂E(η) = min ∂η
3
W (x ; c ; n) =
i=1
⎧ |xi| ≤ ci ⎪0 Wi (xi ; ci ; n) = ⎨ n ⎪ ⎩(|xi| − ci) |xi| > ci
(2)
(Ψ|H |Ψ) (Ψ|Ψ)
(3)
where instead of the conventional scalar product of two wave functions Ψ1 and Ψ2, the bilinear form (Ψ|Ψ 1 2) = ⟨Ψ*1|Ψ2⟩
(4)
is employed. In CODFT, eq 3 is transformed23 into a density functional version by means of the Levy constraint search procedure30 ⎛ E = stat⎜ ρ ⎝
W → PWP =
⎞ ρ (r)ρ(r′) + E XC[ρ , N ]⎟ ⎠ 2|r − r′|
3
k
k
∑
∑
i = iHOMO j = jHOMO
|ψĩ )(ψĩ |W |ψj̃ )(ψj̃ | (7)
where k is the total number of the orbitals. Note that the projection in eq 7 employs the bilinear form defined in eq 4. The projection operator P is updated in each cycle of the SCF procedure. The self-consistent Hartree−Fock (HF) or KS problem is solved for a series of η values. The resulting η trajectory (E(η)) is then examined to identify the η that satisfies the stability condition in eq 2. Not completely unexpected, the convergence of the self-consistent calculations is not easy to achieve. There are several factors that contribute to this problem. First, we are describing systems that feature unbound electrons, where changes in the wave function might only have little impact on the energy. Second, the quantities of interest (orbitals and energies) are complex, implying that the number of parameters is doubled compared to conventional calculations. To cope with the convergence problems, we employ damping, that is, the density matrix of a given iteration is mixed with the density matrix of the previous iteration. Furthermore, the converged density matrices for given η values are used as initial guesses in calculations with similar η values. To examine its performances, we apply the CAP-HF and CAP-CODFT approaches to the 2P state of Be− and the 2Π state of N−2 . For Be−, the lowest 2P resonance is widely studied theoretically as the testing ground for new methods, even if, as of yet, no experimental results are available. There are various publications,1−4,8 applying complex scaling to Be− within HF and CI. We are not aware of CAPs-based calculations for this system, and we employ this approach in conjunction with HF and CODFT to the 2P resonance state of Be−. In calculations using CAPs, it is important that the Gaussian basis set be flexible enough to describe the wave function in the
∫ d r ρ(r)ν(r) + TS[ρ , N ] + ∫ d r d r′ 3
(6)
Here, ci, i = 1−3, are real and non-negative parameters defining the size of a rectangular box. The target atom or molecule is placed in the center of the box. Following ref 17, the matrix elements of W(x;c;n) are calculated within a basis of Cartesian Gaussians, and they are added to the Fock and KS matrices, respectively. The work of Katriel and Davidson and others (see, e.g., refs 32 and 33) shows that asymptotically, as one electron coordinate wanders away from the finite system, the wave function collapses to a single orbital times the ground-state wave function of the (N − 1)-electron system, which decays exponentially fast compared to the single orbital. Therefore, for η → 0, where the complex potential acts only at infinite distance from the system, it impacts only the single outermost electron. In a real calculation, the CAP is applied at a finite distance, and it does act on all of the orbitals to a varying extent. To remedy this problem, the matrix representation of the CAP in eq 6 is not directly added to the KS or Fock matrix; rather, we perform a projection of the potential onto the HOMO and all of the unoccupied molecular orbitals. The projected CAP is
Because the Hellmann−Feynman theorem is satisfied, eq 2 yields the absolute value of Γ/2. The condition in eq 2 is a purely technical one (mandated by the finite basis set), and there is no obvious physical interpretation of it. To meet it, the energy has to be known as a function of η. To obtain E(η), we employ CODFT, which is derived from the energy expression12,13 E[Ψ] =
∑ Wi (xi ; ci ; n)
3
(5)
where ρ is a complex-valued generalization of the electron density calculated according to ρ(r) = (ψ|ρ̂(r)|ψ)/(ψ|ψ). ρ̂(r) = ∑i=1,...,N δ(ri − r) is the electron density operator of an Nelectron system whose complex external potential is ν(r). TS[ρ,N] is the noninteracting kinetic energy functional, and EXC[ρ,N] is the exchange−correlation energy functional of CODFT.23 As indicated in eq 5, the energy is made stationary with respect to variations of ρ. The definitions of TS[ρ,N] and EXC[ρ,N]23 ensure that the stationary point found is the lowest resonance of the system. Along with CODFT, we implemented complex Hartree− Fock (COHF) in the Gaussian software package.31 COHF is also based on the energy functional in eq 3. The details of this implementation will be discussed in a future publication. The energy expression in eq 5 is made stationary with respect to variations of ρ by means of the conventional KS procedure in which the usual scalar product has been replaced by the bilinear form in eq 4 throughout and also in the calculation of expectation values (for an example, see eq 3). The successful implementation of this procedure is confirmed through the validation of the Hellmann−Feynman theorem. In our case, the CAP is defined as a box potential,17 a form that has proven to be successful in combination with the CI method20 1917
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Table 1. Resonance State of Be− as Described by CAP-HF and CAP-CODFTa
region containing the molecule out to the surrounding CAP. For beryllium, we use the 6-31++G* basis of the Gaussian31 program. Furthermore, an additional sp shell with exponent 0.04 is added. In the case of beryllium and in the case of the nitrogen molecule, we did not attempt to undertake a detailed basis set convergence study. Our primary objective was to establish the feasibility of CODFT calculations yielding complex total energies. The total number of primitive Gaussian functions is 36, grouped as (12s,6p,1d) and contracted to [5s,4p,1d]. Another technical factor that influences the results obtained with CAPs is the box size of the potential defined in eq 6. The box size dependence is analyzed in some detail below, and we find that a moderate box size of 6 Bohr is suitable and subsequently used for the CAP-CODFT calculations for beryllium. The exponent n in the CAP eq 6 is set to n = 2. We performed CODFT calculations using the local spindensity functional approximation to exchange and the local spin-density functional approximation to correlation in the parametrization of Vosko, Wilk, and Nusair.34 Because the density is complex in the present context and because the exchange−correlation functionals contain functions of the complex density, such as powers of the density, a phase convention has to be adopted to completely define the functionals. As in our previous work,23 we restrict the phase of the density, that is, the angle of the density vector in the complex plane φ, to φ ∈ (−π,π]. In Figure 1, the energy
CAP (this work)
complex scaling3
method
ERe (eV)
Γ (eV)
HF ELSD X ELSD XC ELSD post-HF X ELSD XC post-HF HF Slater basis CI singles, doubles CI singles, doubles, triples
1.022 1.123 0.455 1.277 0.580 0.688 0.580 0.323
0.219 0.277 0.230 0.281 0.223 0.510 0.377 0.296
a
In the case of CODFT, the functionals employed in the calculation of the resonances are listed. For comparison, results from the literature are added in the lower part of the table.
store the corresponding converged CAP-HF density matrix. The so-obtained η value and density matrix are then used to perform non-self-consistent KS calculations within the local approximation to exchange and correlation. As can be seen in Table 1, the post-HF calculations give resonance positions and lifetimes close to those of the self-consistent CAP-CODFT. Contrary to complex scaling HF, where neither Gaussian1 nor the Slater basis set3 yields a clear cusp in the η trajectory of Be−, our implementation of CAPs exhibits a cusp. As already mentioned above, finite basis sets and finite box sizes are employed, and the associated choices affect the results obtained. We present the dependence on the box size of our CAPs implementation in Figure 2. While increasing the box size
Figure 1. The η trajectories of the 2P resonance state of Be− calculated LSD by CAP-HF, CAP-ELSD X , and CAP-EXC . The insets zoom in on the cusps, where the x-axis is stretched by a factor of about 10. The energy unit is Hartree.
Figure 2. The η trajectories for the resonant state of Be−. The trajectories are obtained with CAP-HF for box sizes ranging from 2 to 12 Bohr. The energy unit is Hartree.
eigenvalues obtained with the CAPs are shown as a function of η. The cusps in the η trajectory indicate resonance states. To obtain the resonance position and width, the ground-state energy of the neutral beryllium atom is calculated and subtracted from the complex energy of the resonance. Table 1 lists the results of our CAP calculations as well as the results3 obtained with the complex scaling method. CAP-CODFT in the local exchange approximation is in reasonable agreement with CAP-HF. As expected, the local approximation underbinds the outer electron. This shortcoming is partially remedied with the local exchange−correlation approximation. The latter stabilizes the system compared to local exchange and yields results that are similar to the complex-scaled CI calculations. For later use, we also introduce a post-HF procedure, where we determine the η value that yields the cusp with CAP-HF and
for a fixed basis set, the cusp becomes sharper, and the imaginary part of the energy approaches zero. This is not surprising because moving the CAP out to infinity for a finite basis set effectively eliminates this potential. Reducing the box size results in a CAP that acts inside of the system, that is, the CAP is nonzero in regions where the physical potential (nuclear attraction) is sizable. This modifies the nature of the system and leads to spurious results. The appropriate box size value (=6 Bohr) lies between these extremes. However, there is a degree of uncertainty associated with this procedure. This uncertainty originates from the CAPs approach and not from our implementation of it in HF and CODFT. 1918
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The resonance state 2Πg of N−2 is well-studied with the CAPs approach20,21,35 and the complex scaling method,5,7 and the corresponding results can be compared to the ones obtained with the present implementation. For N−2 , CAP-CODFT diverges, and we only obtain convergence with the CAP-HF. However, we employ the converged CAP-HF density matrix and the associated η value and perform post-HF CAP-CODFT calculations. The technical parameters of the N−2 calculations are as follows. The bond distance between the nitrogen atoms is 2.069 Bohr, the same as the one employed in ref 16. The N-atom basis set is from ref 36, with 88 Cartesian Gaussians contracted to 58 basis functions. This basis set is similar to the one of ref 21, which contains also additional diffuse p functions. As in the case of beryllium, there is some degree of uncertainty associated with the determination of the appropriate box size. Because of the similarity of our basis set and the one of ref 21, we also employ the rectangular box of this work. We verify that this box size is appropriate by calculating the η trajectories for various sizes, as was done in Figure 2. In Table 2, the position and width of the N−2 resonance as obtained by CAP-HF and CAP-CODFT are provided and
approach37 for the calculation of the ionization potential. The ΔSCF approach is known to yield realistic results for the ionization potentials of stable systems, and we demonstrated that this is also the case for metastable systems described within CAP-CODFT. It might also be possible to extract the resonance position and lifetime directly from the orbital energies. While this appears to be the case within the method of Wasserman and co-workers,38 this question has not yet been addressed within CODFT, and no physical interpretation of the orbital energies is available yet. The extension to metastable systems adds a new line of research to the field of DFT, and we demonstrate that realistic problems can be tackled by CODFT.
■
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Y.Z. would like to acknowledge stimulating discussions with Xiaogang Wang. The authors also thank Philippe Rocheleau for a critical reading of the manuscript. This work is supported by NSERC.
Table 2. Resonance State of N−2 Described by CAP-HF and CAP-CODFTa CAP (this work)
experiment20
method
ERe (eV)
Γ (eV)
HF post-HF ELSD X ELSD XC post-HF complex scaling HF5 CAP-CI36 CAP-FSMRCC21
3.28 4.64 3.39 3.19 2.97 2.520 2.32
0.395 0.532 0.506 0.44 0.65 0.390 0.41
AUTHOR INFORMATION
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REFERENCES
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a
In the case of CODFT, the functionals employed in the calculation of the resonances are listed. For comparison, results from the literature are added in the lower part of the table.
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