Charge transfer, double and bond-breaking excitations with time-dependent density matrix functional theory

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1 Version Charge transfer, double and bond-breaking excitations with time-dependent density matrix functional theory K.J.H. Giesbertz, E. J. Baerends, and... Afdeling Theoretische Chemie, Vrije Universiteit, De Boelelaan 183, 181 HV Amsterdam, The Netherlands (Dated: April 3, 28) Time-dependent density functional theory in its current adiabatic implementations exhibits three remarkable failures: a) totally wrong behavior of the excited state surface along a bond-breaking coordinate; b) lack of doubly excited configurations, affecting again excited state surfaces; c) much too low charge transfer excitation energies. These TDDFT failure cases are all strikingly exhibited by prototype two-electron systems such as dissociating H 2 (zero σ g σ u excitation energy at R, missing lowest excited state (a double excitation) at R > 2.5 bohr) and HeH + (charge transfer excitations). The exact density matrix functional being known in two-electron systems, the performance of time-dependent density matrix theory can be tested in those systems. We find for TDDMFT that: a) The first order (linear response) formulation does not prohibit the correct representation of doubly excited states; b) Within previously formulated simple adiabatic approximations the bonding-to-antibonding excited state surface as well as charge transfer excitations are described without problems, but not the double excitations; c) An adiabatic approximation is formulated in which also the double excitations are fully accounted for. I. INTRODUCTION Dissociation of molecular systems poses a serious challenge within the DFT framework. This is already true for the ground state energy curve, cf. e.g. Refs [1, 2]. For excited states the problems are even worse, there are various kinds of excitations which are not correctly represented in TDDFT calculations. Fig. I displays the total failure of TDDFT (in its adiabatic GGA (BP86) variant) for the potential energy surface (PES) of the first excited state of H 2, 1 1 Σ + u. As highlighted in Ref. [3], this PES goes to zero instead of going through a minimum and asymptotically to ca. 1 ev. B3LYP doesn t improve the situation the least. This state corresponds to the simple σ g σ u orbital excitation in H 2. The prototypical photochemical bond breaking event resulting from bonding to antibonding orbital excitation takes place on the triplet surface (1 3 Σ + u ) corresponding to this singlet excited state, hence the great importance of this type of excitation. In Fig. I we show the TDDFT Σ + g excitation energies of dissociating H 2. It is evident that there is an exact state (the third at R e = 1.4 Bohr) which is totally missing in the TDDFT calculations. In the accurate calculations it becomes lower with increasing R and has avoided crossings with the second and first state, becoming the lowest excited 1 Σ + g state (2 1 Σ + g ) from ca. 2.5 Bohr onwards. This is a doubly excited state, (σ g ) 2 (σ u ) 2. It is totally missing in the TDDFT calculations. After some initial optimism that such doubly excited states states would be accurately calculated in TDDFT [4 6], it has become clear this is not the case [7 1]. We note in passing that also in the higher part of the spectrum - above.6 a.u. - TDDFT performs very poorly, there is hardly any agreement with the exact spectrum. In Fig. I the excitation energies of HeH + are shown along the dissociation coordinate for the lowest three 1 Σ excited states. The TDDFT excitation energies exhibit the well-known severe underestimation at long distance, where these excitations have strong charge transfer character, from He(1s) to H(1s) in 2Σ and from He(1s) to H(2s, 2p z ) in 3Σ and 4Σ. The hybrid functional B3LYP improves only little on the pure GGA BP86. We note that the excited state energy does not have a 1/R asymptotic behavior, the fragments being He + and neutral H. A remedy for these deficiencies of time-dependent density functional theory might be sought in density matrix functional theory (DMFT) [11 18]. For the ground state total energy curve along the dissociation coordinate, successes have already been reported [16] with approximate DM functionals that represent improvements on the forms formulated long ago by Müller [11] and Buijse and Baerends [12, 13]. It is the purpose of the present paper to address the abovementioned problem cases for TDDFT excitation energy calculation with DMFT, using a further development of linear response based timedependent density matrix functional theory [19, 2]. We will restrict ourselves to the two-electron systems H 2 and HeH +, which exhibit all the problems, but which have the advantage that we do not have to introduce an approximate functional since the exact one is known for two-electron systems. The spatial part of a singlet twoelectron wave function is symmetric at all times t, so the matrix C(t) of expansion coeeficients in a two-electron product basis χ p (r)χ q (r ) is symmetric and can be diagonalised at each t, as originally noted by Löwdin and Shull [21] Ψ(r, r ; t) = pq = µ C pq (t)χ p (r)χ q (r ) c µ (t)φ µ (rt)φ µ (r t). (1) The orbitals {φ µ } are time-dependent, since the wave function will have a different spectral representation

2 Σ + u 3Σ + u.5 3Σ + 4Σ + 1Σ + u 2Σ + R He-H.8 4Σ + g FIG. 2: Σ + TDDFT excitation energies for HeH +. Solid lines: exact in cc-pvtz basis; dashed: TDDFT-BP86; dotted: TDDFT-B3LYP Σ + g 3Σ + g FIG. 1: Σ + u and Σ + g TDDFT excitation energies for H 2. Solid lines: exact in cc-pvtz basis; dashed: TDDFT-BP86; dotted: TDDFT-B3LYP. matrix reduces to the particularly simple form Γ(r 1, r 2, r 2, r 1; t) = pqrs Γ pqrs (t)χ s(r 1)χ r(r 2)χ q (r 2 )χ p (r 1 ) = µνρσ[ 2cµ (t)c σ(t)δ µν δ στ ] φ σ(r 1t)φ ρ(r 2t)φ ν (r 2 t)φ µ (r 1 t). (3) This shows that in a singlet two-electron system the twomatrix can be reconstructed easily from the one-matrix. It is this property which gives two-electron systems their special place in DMFT. The electron-electron interaction energy W (t) = Ψ(t) Ŵ Ψ(t) can be written in terms of the matrix W W µν (t) ρστ Γ µρστ (t)w τσρν (t) at different times. To distinguish this special timedependent basis {φ µ (t)} from the stationary one, {χ p }, we use Greek letters as indices for the time-dependent basis and Latin for the time-independent functions. The one-particle density matrix can be written = 2 c σ(t)c µ (t)w σσµν (t) (4) σ w pqrs = dr 1 dr 2 χ p(r 1 )χ q(r 2 )w(r 1, r 2 ) χ r (r 2 )χ s (r 1 ) (5) γ(r, r ; t) = pq = µ γ pq (t)χ p (r)χ q(r ) n µ (t)φ µ (rt)φ µ(r t) (2) as W (t) = 1 2Tr{W }. Note that the two-electron integrals are related to the ones in conventional notation as w pqrs = pq sr. The equation of motion of the onematrix can now be written in the form that was derived in [19] and used in [2] which shows the {φ µ (t)} are the time-dependent NOs and the wavefunction coefficients c µ (t) of Eq. 1 are related to the NO occupation numbers, n µ (t) = 2 c µ (t) 2. The two- i γ kl (t) = ( hks (t)γ sl (t) γ ks (t)h sl (t) ) + s ( W kl (t) W kl(t) ), (6)

3 3 where W kl (t) is defined as the contraction of the twomatrix in the stationary basis with the two-electron integrals in the same basis, W kl (t) = rst Γ krst(t)w tsrl (t). The simplifications afforded by the simple representation of the one-matrix and two-matrix in terms of the diagonal wavefunction coefficients {c µ (t)}, can now be exploited to derive (with considerable algebra, see Ref. ) exact equations for the excitation energies. We first define the matrices A kl,sr (δ kr δ ls + δ ks δ lr )E (w klsr + w klrs ) (δ kr h ls + δ ks h lr + δ lr h ks + δ ls h kr ) (7a) C kl,r A kl,rr (7b) E k,r A kk,rr, (7c) where E is the energy of the ground state. Note that A is hermitian, and C and E are just submatrices of A. In order to obtain a symmetric matrix problem, we define quantities related to the real and imaginary parts of the lower triangle density matrix elements δγ kl, l < k and the real diagonal elements δγ kk = δn κ : δ γ R/I kl () δγr/i kl () 2(c l ± c k ) (8) We then obtain the equations CC T A CE A C δ γr () δ γ I () = δ γr () δ γ I () (9) EC T C T E2 δñ() δñ(). The values for which this system gives non-trivial solutions represent excitation energies. The equations have to be solved iteratively, e.g. by diagonalizing the matrix for trial values of, and search for those values of where an eigenvalue ɛ i () becomes equal to the input. Beyond the linear response approach, no approximations have yet been introduced, and indeed the exact excitation energies are found, equal to those form a full CI calculation in the same basis set. This demonstrates that the linear response formulation does not preclude obtaining double excitations. Het berekenen van excitatie enrgieën met lineaire response is geen benadering, dus ik zou gewoon zeggen dat er geen benaderingen zijn gemaakt. For practical applications (systems with more than two electrons), we need adiabatic approximations, which yield an -independent matrix so that simple diagonalisation will yield the excitation energies. The limit of the exact TDDMFT matrix of Eq. 9 is relevant for the static limit of frequency dependent polarizability calculations. It is, however, immediately clear that the 1/ terms in the exact Eq. 9 will diverge for small. Not surprisingly, the so-called static approximation (SA) introduced in Ref. [19] yields a matrix in which those terms have been eliminated: A A C δ γr () δ γ I () = δ γr () δ γ I () (1) C T δñ() δñ() Σ + u Σ + g FIG. 3: Σ + u and Σ + g excitations for H 2. Solid lines: exact and AA2. Dotted: SA. Dashed: AA1. Unfortunately, the SA approximation turns out to be deficient in the sense that it yields m excitation energies equal zero. As we will see, it is precisely the doubly excited states that turn out to be totally wrong (there are m diagonal excitations (ψ 1 ) 2 (ψ i ) 2 ); ψ 1 is the occupied 1σ g or 1σ orbital in our systems). In an attempt to improve upon the SA, the so-called AA has been proposed [2], in which the third set of equations of the SA has been replaced by the static perturbation equation for the energy under a (static) potential perturbation, C T δ γ R () + Eδñ() δṽ D () = c δe(), (11) Again, this approximation, which we will call AA1, does not solve the problem with the double excitations, which are actually totally absent in this approximation. We propose an adiabatic approximation in which the static perturbation equation 11 is incorporated in the third set of the equations for the (δ γ R, δ γ I, δñ) vectors, in such

4 Σ + R He-H FIG. 4: Σ + excitations for HeH +. Solid lines: exact and AA2. Dotted: SA. Dashed: AA1. a way that in the limit these equations reduce to the static perturbation equation. In that way the full set of equations becomes consistent, because imposing the equation 11 is actually a requirement for the elimination of the 1/ terms in the first two sets of equations. We obtain the AA2 set of equations A A C δ γr () δ γ I () = δ γr () δ γ I () (12) C T C T E δñ() δñ(). We compare in Fig. I the adiabatic TDDMFT excitation energies with the exact ones for the Σ + u excited states. Interestingly, in this symmetry the SA, AA1 and AA2 all coincide with the exact excitation energies along the whole binding distance. This is related to the symmetry: we find that for different symmetries than the ground state, all adiabatic approximations yield perfect excitation energies. This also holds for the Π u and Π g excited states. The significant observation, however, is that the adiabatic TDDMFT, in contrast to adiabatic TDDFT, has no problem with the σ g σ u excitation along the whole bond-breaking coordinate. Results for the Σ + g excitations are shown in Fig. I. Now the adiabatic approximations are not identical to the exact ones. However, the performance of the adiabatic approximations SA and AA1 is rather good, except for the fact that they are completely missing the state with predominantly doubly excited character, that is high-lying at short distance and then on its way down crosses the other states (see the crosses indicating the doubly excited electronic character on the different curves). This deficiency of the SA and AA1 adiabatic approximations in TDDMFT is actually very similar to the adiabatic TDDFT deficiency discussed in the Introduction. On the other hand, the AA2 approximation coincides with the exact results. It brings in the coupling to the diagonal doubles very effectively, in fact perfectly. Finally, the HeH + excitations in the Σ + symmetry are shown in Fig I. The low lying Σ + excitations are, similar to the Σ + g excitations in H 2, not perfect in the SA and AA1 approximations, but they are clearly rather accurate. In this case the AA1 performs better than the SA, especially for the lowest Σ + excitation and also for the other two. The AA2 is exact for all the Σ (and Π) excited states. The important point is that at long distance the lowest excited state, 2Σ + at ca..4 a.u., is a pure CT state, He(1s) H(1s). On the other hand, 3Σ + at ca..74 a.u. is at long distance a mixed local (He(1s) He(2s)) and charge transfer (He(1s) H(2s, 2p)) excitation. The 4Σ + at ca..82 a.u. is mostly CT, He(1s) H(2s, 2p). The different characters of these states do not seem to influence the accuracy of the adiabatic TD- DMFT approximations. We conclude that the well-known failure cases of TDDFT can in principle be solved without difficulty in adiabatic TDDMFT. Whether adiabatic TDDMFT will prove to be widely applicable, also in N-electron systems with N > 2, will depend on the quality of the approximate density matrix functionals that are needed in those cases. Good progress is being made with such functionals for the ground state, both at equilibrium geometry [22] and along the dissociation coordinate,[16] which offers good prospects for the response calculations. [1] E. J. Baerends, Phys. Rev. Lett. 87, 1334 (21). [2] M. Grüning, O. V. Gritsenko and E.J. Baerends, J. Chem. Phys. 118, 7183 (23). [3] O. Gritsenko, S. J. A. van Gisbergen, A. Görling and E.J. Baerends, J. Chem. Phys. 113, 8478 (2). [4] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 314, 291 (1999). [5] C-P Hsu, S. Hirata and M. Head-Gordon, J. Phys. Chem. A 15, 451 (21). [6] D.J. Tozer, R.D. Amos, N.C. Handy, B.J. Roos and L. Serrano-Andres, Mol. Phys. 97, 859 (1999). [7] N.T. Maitra, F. Zhang, R.J. Cave and K. Burke, J. Chem. Phys. 12, 5932 (24). [8] R.J. Cave, F. Zhang, N.T. Maitra and K. Burke, Chem. Phys. Lett. 389, 39 (24). [9] J. Neugebauer and E.J. Baerends, J. Chem. Phys. 121, 6155 (24). [1] I.A. Mikhailov, Phys. Rev. A 77, 1251 (28). [11] A.M.K. Müller, Phys. Lett. A 19, 446 (1984). [12] M. Buijse, Ph.D. thesis, Vrije Universiteit, Amsterdam (1991). [13] M. Buijse and E.J. Baerends, Mol. Phys. 1, 41 (22).

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