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Charge-transfer (CT) orbitals for the one-electron description of CT excitations in a wide range of donor-acceptor separations
Chemical Physics Letters 667 (2017) 51–54
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Charge-transfer (CT) orbitals for the one-electron description of CT excitations in a wide range of donor-acceptor separations O.V. Gritsenko Section Theoretical Chemistry, VU University, Amsterdam, The Netherlands
a r t i c l e
i n f o
Article history: Received 6 October 2016 In final form 17 November 2016 Available online 19 November 2016
a b s t r a c t A transformation of the virtual Kohn-Sham orbitals is proposed to a set of charge-transfer orbitals (CTOs) adapted to description of CT excitations. The CTO scheme offers a simple estimate of the CT excitation energy with an orbital energy difference. This estimate reproduces well the reference values of the configuration interaction (CI) method in a wide range of donor-acceptor separations in the paradigmatic He Be complex. CTO-based orbital energy and shape indices are proposed to assess the suitability of the CT description with virtual orbitals of a given basis set. Both indices yield correct trends for the Kohn-Sham and Hartree-Fock orbitals. Ó 2016 Elsevier B.V. All rights reserved.
Different physical nature of excitations in compact molecules (and intrafragmental excitations in larger systems), on the one hand, and of long-range charge-transfer (CT) excitations between an electron donor (D) and electron acceptor (A), on the other hand, greatly affects their one-electron description. As was emphasized in the literature [1,2], the Kohn-Sham (KS) orbitals of density functional theory (DFT) are especially suitable for the former type of excitations. As a rule, localized excitations are efficiently described as (a combination of a few) single orbital excitations /i ! /a from the occupied KS orbital /i ðrÞ to the virtual orbital /a ðrÞ. Accordingly, the energies xa of predominantly single excitations are fairly approximated with the differences of the orbital energies
xa xTDDFTð0Þ ¼ a i : a
ð1Þ
In time-dependent DFT (TDDFT) [3–7] the zero-order estimate (1) is often further improved by the coupling of individual orbital excitations via the Hartree-exchange-correlation (Hxc) kernel f Hxc ðr 1 ; r 2 ; xÞ. As was established recently [8], a similar efficient description of localized excitations can be achieved in TD Hartree-Fock (TDHF) [9–15] and density matrix functional (TDDMFT) [16–19] theories by the transformation of the HF and natural (NO) orbitals, respectively, to the natural excitation orbitals (NEOs) ha . The latter orbitals are obtained with the diagonalization of the submatrix of the (inverse) response matrix of the TDHF/TDDMFT eigenvalue problem for electronic excitations. This diagonalization is performed in the subspace of the virtual HF orbitals or weakly occupied NOs and the resultant NEOs resemble the virtual KS orbitals. The physical reason behind the established trend is that both virtual KS
orbitals and NEOs represent an electron in the field of N electrons plus an effective hole of the xc origin surrounding the electron, ðN 1Þ electrons in total. For localized excitations this hole efficiently simulates the hole in an excited state left behind by the excited electron. Previously, model effective potentials of ðN 1Þ electrons were employed within the improved virtual orbital (IVO) [20–24] procedure to improve the description of excitations with singly excited electronic configurations. Unfortunately, because of the very same physical reason, the virtual KS orbitals are not suitable for the description of the long-range CT excitations D : A ! Dþ A in the D A complexes [1,25–28]. At the large D A separation R the true CT excitation energy xCT turns to the difference between the vertical ionization potential (VIP) ID of D and the electron affinity A A of A corrected with the electron-hole interaction 1=R
xCT ID A A 1=R:
ð2Þ
The difference ðID A A Þ, which dominates at large R, is exactly rep-
D resented in the rigorous KS theory, ID A A ¼ A S H , with that A D between the energies S and H of the singly occupied molecular
orbital (SOMO) /A S of the anionic A fragment and highest occupied
MO (HOMO) /DH of the D fragment. The former energy is substantially higher than that LA of the lowest unoccupied MO (LUMO) of
A the neutral A, the corresponding positive difference D A ¼ A S L is often called the derivative discontinuity [26,29,30]. This holds true since, as was stated above, an electron on the virtual KS orbital
of the neutral A effectively interacts with ðN A 1Þ electrons, while E-mail address: [email protected] http://dx.doi.org/10.1016/j.cplett.2016.11.034 0009-2614/Ó 2016 Elsevier B.V. All rights reserved.
an electron on the KS orbital of the anion A interacts with N A
52
O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
electrons. Because of this, the zero-order TDDFT (1) yields too low estimate of xCT [25]
suitable for the description of CT excitations. The transformation is performed by diagonalizing in the subspace f/a g
TDDFTð0Þ xCT xCT ID A A þ DH LA ¼ D A :
hs
ð3Þ
As was established in [27,30], in order to correct xCT the Hxc kernel f Hxc should diverge in the long-range limit. However, f Hxc of the conventional DFT functionals lack such a divergence, so that conventional TDDFT significantly underestimates xCT . In this Letter the long-range CT problem of TDDFT is illustrated with the example of the paradigmatic D A complex He Be. We consider the CT excitation associated with an electron transfer from the 1s HOAO of the He atom to the 2pp LUAO of the Be atom. Fig. 1 presents the plot of the reference xCISDT values calculated at CT various separations R(He Be) with the configuration interaction (CI) method with all single, double, and triple excitations (CISDT) from two valence MOs and with the 1r2 core frozen. The calculations were performed with the GAMESS (US) package of programs [31] in the aug-cc-pVTZ basis (without the f-functions) [32,33]. The excitation energy xCISDT exhibits the minimum at R = 2.0 Å and its CT further increase amounts to 3.11 eV at 8.0 Å. obtained with the standard Full-linear-response TDDFT xBLYP CT combination BLYP of the exchange functional by Becke (B88) [34] and the correlation functional by Lee, Yang, and Parr (LYP) [35,36] as well as with the corresponding Hxc kernel displays a qualitatively wrong trend (see Fig. 1). Being consistently much lower than xCISDT , the energy xBLYP decreases monotonically with CT CT R, so that the TDDFT-BLYP error increases (in absolute value) from 3.41 eV at R = 1.5 Å to 9.23 eV at R = 8.0 Å. Furthermore, full-linearresponse time-dependent generalized KS (TDGKS) approach with the B3LYP [37] and M06 [38] hybrid functionals also produces substantially underestimated xCT with the deficient dependence on R. Both hybrid functionals yield a displaced and too shallow minimum at R = 3.5 Å, with the absolute xCT error increasing from 3.18 eV at R = 2.0 Å to 7.01 eV at R = 8.0 Å for TDDFT-B3LYP and from 2.17 eV at R = 1.5 Å to 6.64 eV at R = 8.0 Å for TDDFT-M06. In this Letter we introduce a set of CT orbitals (CTOs) fna g, which are obtained with a transformation of the ground-state virtual KS orbitals f/a g of a D A complex and which would be TDDFTð0Þ
PCT" " na
" ¼ n" a na
ð4Þ
^s ðrÞ of D A augof the conventional KS operator h ^ mented with the projected CT (PCT) correction hPCT" ðrÞ the matrix
PCT" hs
^PCT" ðrÞ ¼ h ^s ðrÞ þ h ^PCT" ðrÞ ¼ hðrÞ ^ ^PCT" ðrÞ: h þ v Hxc ð½q; rÞ þ h s
ð5Þ
^ In (5) hðrÞ is the one-electron operator and v Hxc ð½q; rÞ is the Hxc potential of the ground-state density of D A, which consists of ^PCT" reads the Hartree v H and xc v xc parts. The PCT correction h
^PCT" ðrÞ ¼ P b A ½v " ð½qCT ; rÞ v Hxc ð½q; rÞ P bA: h Hxc
ð6Þ
Here, v q ; rÞ is the Hxc spin-up potential of the spinunrestricted KS approach calculated with the density qCT ðrÞ " Hxc ð½
CT
qCT ðrÞ ¼ qðrÞ
X X ci j/"iðDÞ ðrÞj2 þ ca j/"aðAÞ ðrÞj2 ; i2CT
ð7Þ
a2CT
which (with the properly chosen coefficients ci and ca ) represents a combined CT from the occupied (up-spin-) orbitals /"iðDÞ ðrÞ of D A associated with D to the virtual orbitals /"aðAÞ ðrÞ associated with A. It is desirable to apply the PCT correction only to the virtual orbitals localized on A, which participate in the CT D : A ! Dþ A . This bA is achieved by employing in (6) the projection operator P
bA ¼ P
X
j/pA ih/pA j
ð8Þ
bound
onto the bound KS orbitals /pA of the individual fragment A. We would like to stress the following simplifications and virtues of the proposed CTO scheme (4)–(8) compared to the DSCFDFT scheme [39,40], in which the initial qCT of (7) is optimized with the full self-consistent field (SCF) DFT calculation: (1) The transformation of the KS orbitals f/a g to the CTOs fna g amounts to only one cycle of the SCF procedure with a single diagonalization (4).
Fig. 1. The CT excitation energy as a function of R(He Be) calculated with CISDT, CTO, and TDDFT.
53
O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
(2) More importantly, the CTO scheme is free from the convergence problems of DSCF-DFT associated with a possible variational collapse to a lower excited configuration as well as to the ground state [41]. During the CTO construction, qCT is not updated. Also, the resultant CTOs are composed from the virtual KS orbitals, so they are orthogonal to the occupied KS orbitals /i , while the latter remain intact. All this makes the abovementioned posibility of the collapse irrelevant, so there is no need in the imposition of constraints, which enforce orbital transitions in the constricted variational DFT (CV-DFT) [42]. The outlined stability of the CTO scheme allows, in principle, to broadly vary the form and composition of qCT to obtain more suitable CTOs in analogy to a variation of the modified one-electron HF operator in the abovementioned IVO procedure [20–24]. Compared to TDHF, the advantage of the CTO scheme is the inclusion of the effect of electron correlation on the calculated orbital energies through the use of some approximate correlation potential within v xc . (3) Last, but not the least, the CTO scheme exploits the full potentiality of one-electron theory. While in DSCF-DFT the excitation energy xCT is calculated as the difference between qCT and the total energies corresponding to DSCFDFT CT q; xCT ¼ E½q E½q, in the CTO scheme xCT is estimated simply as the difference between the orbital energies " n" a and i of the corresponding CTO na and the occupied KS orbital /i n" xCTO a ¼ a i :
ð9Þ
Of course, the CTO estimate (9) requires to use an xc potential v xc , which would yield a good quality orbital energies [43] (see also the presentation below). In its turn, the shape of the CTO represents the density distribution of the excited electron. Could the proposed CTO scheme provide an adequate oneelectron description of the abovementioned paradigmatic CT in the He Be complex? To address this question, the CTOs n"a and the CTO energies n" a are obtained at various R(He Be) separations. The He Be ground state is calculated with the following approximate xc potential
v xcBVWNðSPPÞ
3 2
B VWNðSPPÞ v xcBVWNðSPPÞ ðrÞ ¼ v LDA ðrÞ: x ðrÞ þ 2x ðrÞ þ v c
ð10Þ
where v LDA is the exchange potential of the local density approxix mation (LDA), Bx ðrÞ is the B88 exchange energy density [34], and
v VWNðSPPÞ is the LDA correlation potential of Vosko, Wilk, and Nusair c
(VWN) [44] with the correction of Stoll, Pavlidou, and Preuss (SPP) [45]. The first two terms in (10) approximate the Slater’s exchange potential which, as was shown in [43], yields (when augmented with a suitable correlation potential) good quality orbital energies. The potential (10) is chosen, since its 2s HOAO energy of the Be atom
BVWNðSPPÞ 2s ¼ 9:37 eV perfectly reproduces (as is required in
the KS theory [46–49]) the (minus) first VIP of Be, IBe 1 ¼ 9:32 eV. Fur-
thermore, its 1s energy of the He atom 1s
BVWNðSPPÞ
¼ 25:15 eV fairly
reproduces the (minus) first VIP of He, IHe 1 ¼ 24:59 eV with the rel-
atively small error D1s
BVWNðSPPÞ
¼ 0:56 eV. The PCT correction (6)
of the spinis obtained with the spin-up component v xc unrestricted version of the potential (10) calculated with the following qCT BVWNðSPPÞ"
qCT ðrÞ ¼ qðrÞ j/"2r ðrÞj2 þ 0:5j/"1px ðrÞj2 þ 0:5j/"1py ðrÞj2 :
ð11Þ
The CTO calculations are performed with the original program [50] adapted to GAMESS-US. The data in Fig. 1 and in Tables 1 and 2 gives an affirmative answer to the question posed in the beginning of the previous paragraph. Fig. 1 displays the CTO estimate xCTO CT of xCT BVWNðSPPÞ n" xCTO : CT ¼ 1p 2r þ D1s
ð12Þ
" Here, n" 1p is the energy of the lowest CTO n1p of the p-symmetry localized on Be and 2r is the B-VWN(SPP) energy of the groundstate occupied KS MO /2r associated with the HOAO of He. In (12) it is corrected with the abovementioned B-VWN(SPP) deviation BVWNðSPPÞ D1s . One can see from Fig. 1, that the CTO estimate (12) greatly reduces the large error of conventional TDDFT. Remarkably, CISDT xCTO in the whole range of R CT reproduces well the reference xCT (He Be). Being consistently higher (in average, by 0.77 eV), xCTO CT ðRÞ reproduces very well the dependence of the reference xCISDT on R, with the error of the difference between xCTO CT CT ð8:0Þ and xCTO ð2:0Þ being of only 0.27 eV. CT Based on the CTO scheme (4)–(9), we propose two orbital indices to assess the efficiency of the CT description with the relevant virtual orbital wa of a given basis. The charge-transfer orbital is given with the difference energy index (CTOEI) ICTE a
ICTE ¼ wa n" a a
ð13Þ
between the orbital energies n"a .
wa
of wa and
n" a
of the corresponding
The smaller is the absolute CTOEI value jICTE CTO a j, the better tends to be the description of the CT energy effect with the zeroorder (TD) DFT estimate (1) with wa . The sign of the CTOEI indicates whether (1) tends to overestimate or underestimate xCT . In its turn, the charge-transfer orbital shape index (CTOSI) ICTS is evaluated a CTOðaÞ
with the coefficient caðwÞ
of wa in the expansion of n"a in the consid-
ered basis fwp g CTOðaÞ
ICTS ¼ jcaðwÞ j2 : a
ð14Þ
The larger the CTOSI value is, the better the shape of wa tends to describe the distribution of the excited electron. Tables 1 and 2 compare the variation with R(He Be) of the CTOEI and CTOSI evaluated for the relevant (for the considered CT) B-VWN(SPP) KS and HF 1p-orbitals. According to both CTOEI and CTOSI, the description of the CT with the KS orbital /1p gradually deteriorates with R(He Be) (See Table 1). Indeed, at shorter distance R = 1.5 Å the CTOEI is close to zero, while the CTOSI practically reaches the maximal unit value. The former monotonically increases in absolute value, while the latter decreases with R (He Be), so at R = 8.0 Å the absolute CTOEI value amounts to 4.88 eV, while the CTOSI drops to 0.424. Table 2 displays the opposite trend for the HF orbital v1p . According to both CTOEI and CTOSI, the description of the CT with v1p gradually improves with R(He Be). Indeed, at R = 1.5 Å the STOEI amounts to 6.18 eV, while the CTOSI is only 0.174. The former monotonically decreases, while the latter increases with R (He Be), so at R = 8.0 Å the CTOEI drops to 1.20 eV, while the CTOSI reaches 0.848. The established variation of the proposed CTOEI and CTOSI with R(He Be) makes a qualitative physical sense. Indeed, at shorter R the complex He Be is more similar to compact molecules for which, as was mentioned in the beginning of this Letter, the KS orbitals provide a fair description of excitations. Unlike to this, in the case of long-range CT the excited electron at the affinity level of A is better represented, by virtue of the Koopmans’ theorem, with a HF MO [51,52]. Summarizing, we have proposed a transformation of the virtual KS orbitals to an orbital set (the CTOs) adapted to description of CT excitations. CTOs and their energies are obtained with a single
54
O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
Table 1 Variation of the charge-transfer orbital energy (CTOEI, in eV) and shape (CTOSI) indices for the B-VWN(SPP) KS virtual orbital /1p with the He Be separation (in Å). CT orbital indexes/R(He Be)
1.5
2.0
3.5
5.0
1p
5.80
4.36
2.36
1.50
1.08
0.81
5.97 0.17 1.000
5.72 1.36 0.985
5.72 3.36 0.792
5.69 4.19 0.598
5.69 4.61 0.473
5.69 4.88 0.424
n" 1p
CTOEI CTOSI
Table 2 Variation of the charge-transfer orbital energy (CTOEI, in eV) and shape (CTOSI) indices for the HF virtual orbital CT orbital indexes/R(He Be)
n" 1p HF 1p
CTOEI CTOSI
6.5
8.0
v1p with the He Be separation (in Å).
1.5
2.0
3.5
5.0
6.5
8.0
5.80
4.36
2.36
1.50
1.08
0.81
0.38
0.40
0.40
0.39
0.39
0.39
6.18 0.174
4.76 0.214
2.76 0.530
1.89 0.713
1.47 0.813
1.20 0.848
diagonalization of the submatrix of the modified KS operator. The CTO scheme offers a simple estimate of the CT excitation energy with an orbital energy difference. This estimate reproduces well the reference CISDT values in a wide range of the donor-acceptor separations in the paradigmatic He Be complex. At larger R this improvement over the conventional TDDFT is due to the proposed PCT correction, while at the short R it is due to the use of the sufficiently close B-VWN(SPP) approximation to the true KS potential. The CTO-based indices CTOEI and CTOSI have been proposed to assess the suitability of the CT description with virtual orbitals of a given basis set. Both indexes yield the correct trends for the KS and HF orbitals relevant to the considered CT in the He Be. Evert Jan Baerends and Robert van Meer are gratefully acknowledged for critical and stimulating comments. Robert van Meer is also acknowledged for his guidance and assistance in producing and processing of computational data.
[13] [14] [15] [16] [17]
References
[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]
[1] E.J. Baerends, O.V. Gritsenko, R. van Meer, Phys. Chem. Chem. Phys. 15 (2013) 16408. [2] R. van Meer, O.V. Gritsenko, E.J. Baerends, J. Chem. Theory Comput. 10 (2014) 4432. [3] E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52 (1984) 997. [4] M.E. Casida, in: D.P. Chong (Ed.), Recent Advances in Density-Functional Methods, Part I, World Scientific, Singapore, 1995, p. 155. [5] K. Burke, J. Werschnik, E.K.U. Gross, J. Chem. Phys. 123 (2005) 062206. [6] S. Hirata, M. Head-Gordon, Chem. Phys. Lett. 314 (1999) 291. [7] M.A.L. Marques, N.T. Maitra, F.M.S. Nogueira, E.K.U. Gross, A. Rubio, in: Fundamentals of Time-Dependent Density Functional Theory, Lecture Notes in Physics, vol. 837, Springer, Heidelberg, 2012. [8] R. van Meer, O.V. Gritsenko, E.J. Baerends, Chem. Phys. Lett. 639 (2015) 315. [9] J. Frenkel, Wave Mechanics, Advanced General Theory, Clarendon Press, Oxford, 1934. [10] J.P. Blaizot, G. Ripka, Quantum Theory of Finite Systems, MIT Press, Cambridge, Massachusetts, 1986. [11] D.J. Thouless, The Quantum Mechanics of Many-Body Systems, Academic Press, New York, 1972. [12] A. Dalgarno, G.A. Victor, Proc. R. Soc. Lond. A 291 (1966) 291.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
[51] [52]
A. Kerman, S. Koonin, Ann. Phys. 100 (1976) 332. H.B. Schlegel, S.M. Smith, X. Li, J. Chem. Phys. 126 (2007) 244110. H. Eshuis, G. Balint-Kurti, F. Manby, J. Chem. Phys. 128 (2008) 114113. K. Pernal, O. Gritsenko, E.J. Baerends, Phys. Rev. A 75 (2007) 012506. K. Pernal, K. Giesbertz, O. Gritsenko, E.J. Baerends, J. Chem. Phys. 127 (2007) 214101. K.J.H. Giesbertz, O.V. Gritsenko, E.J. Baerends, Phys. Rev. Lett. 105 (2010) 1. K.J.H. Giesbertz, O. Gritsenko, E.J. Baerends, J. Chem. Phys. 136 (2012) 094104. S. Huzinaga, C. Arnau, Phys. Rev. A 1 (1970) 1285. W.J. Hunt, W.A. Goddard III, Chem. Phys. Lett. 3 (1969) 414. K. Morokuma, S. Iwata, Chem. Phys. Lett. 16 (1972) 192. B.J. Mogensen, S. Rettrup, Int. J. Quant. Chem. 44 (1992) 1045. M.W. Schmidt, E. Hull, T. Windhus, J. Phys. Chem. A 119 (2015) 10408. A. Dreuw, J.L. Weisman, M. Head-Gordon, J. Chem. Phys. 119 (2003) 2943. D.J. Tozer, J. Chem. Phys. 119 (2003) 12697. O. Gritsenko, E.J. Baerends, J. Chem. Phys. 121 (2004) 655. N.T. Maitra, J. Chem. Phys. 144 (2016) 220901. D. Vieiera, K. Capelle, C.A. Ullrich, Phys. Chem. Chem. Phys. 11 (2009) 4647. M. Hellgren, E.K.U. Gross, Phys. Rev. A 85 (2012) 022514. M.W. Schmidt, K.K. Baldridge, J. Boatz, S. Elbert, M. Gordon, J. Jensen, S. Kosekt, N. Matsunaga, K. Nguyen, S. Su, T. Windus, M. Dupuis, J. Montgomery, J. Comput. Chem. 14 (1993) 1347. J.T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. D.E. Woon, T.H. Dunning, J. Chem. Phys. 100 (1994) 2975. A.D. Becke, Phys. Rev. A 38 (1988) 3098. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. B. Miehlich, A. Savin, H. Stoll, H. Preuss, Chem. Phys. Lett. 157 (1989) 200. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. Y. Zhao, D.G. Truhlar, Theor. Chem. Acc. 120 (2008) 215. J.C. Slater, Adv. Quant. Chem. 6 (1972) 1. T. Ziegler, A. Rauk, E.J. Baerends, Theor. Chim. Acta 43 (1977) 261. Y.C. Park, M. Krykunov, T. Ziegler, Mol. Phys. 113 (2015) 1636. T. Ziegler, M. Krykunov, I. Seidu, Y.C. Park, Top. Curr. Chem. 368 (2016) 61. O.V. Gritsenko, L.M. Mentel, E.J. Baerends, J. Chem. Phys. 144 (2016) 204114. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. H. Stoll, M. Pavlidou, H. Preuss, Theor. Chem. Acc. 149 (1978) 143. J.P. Perdew, R.G. Parr, M. Levy, J. Balduz, Phys. Rev. Lett. 49 (1982) 1691. J.P. Perdew, M. Levy, Phys. Rev. Lett. 51 (1983) 1884. M. Levy, J.P. Perdew, V. Sahni, Phys. Rev. A 30 (1984) 2745. C.-O. Almbladh, A.C. Pedroza, Phys. Rev. A 29 (1984) 2322. M. Buijse, Electron Correlation (Ph.D. thesis), Vrije Universiteit, Amsterdam, 1991. T. Tsuneda, J.W. Song, S. Suzuki, K. Hirao, J. Chem. Phys. 133 (2010) 174101. T. Stein, H. Eisenberg, L. Kronik, R. Baer, Phys. Rev. Lett. 105 (2010) 266802.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Charge-transfer (CT) orbitals for the one-electron description of CT excitations in a wide range of donor-acceptor separations O.V. Gritsenko Section Theoretical Chemistry, VU University, Amsterdam, The Netherlands
a r t i c l e
i n f o
Article history: Received 6 October 2016 In final form 17 November 2016 Available online 19 November 2016
a b s t r a c t A transformation of the virtual Kohn-Sham orbitals is proposed to a set of charge-transfer orbitals (CTOs) adapted to description of CT excitations. The CTO scheme offers a simple estimate of the CT excitation energy with an orbital energy difference. This estimate reproduces well the reference values of the configuration interaction (CI) method in a wide range of donor-acceptor separations in the paradigmatic He Be complex. CTO-based orbital energy and shape indices are proposed to assess the suitability of the CT description with virtual orbitals of a given basis set. Both indices yield correct trends for the Kohn-Sham and Hartree-Fock orbitals. Ó 2016 Elsevier B.V. All rights reserved.
Different physical nature of excitations in compact molecules (and intrafragmental excitations in larger systems), on the one hand, and of long-range charge-transfer (CT) excitations between an electron donor (D) and electron acceptor (A), on the other hand, greatly affects their one-electron description. As was emphasized in the literature [1,2], the Kohn-Sham (KS) orbitals of density functional theory (DFT) are especially suitable for the former type of excitations. As a rule, localized excitations are efficiently described as (a combination of a few) single orbital excitations /i ! /a from the occupied KS orbital /i ðrÞ to the virtual orbital /a ðrÞ. Accordingly, the energies xa of predominantly single excitations are fairly approximated with the differences of the orbital energies
xa xTDDFTð0Þ ¼ a i : a
ð1Þ
In time-dependent DFT (TDDFT) [3–7] the zero-order estimate (1) is often further improved by the coupling of individual orbital excitations via the Hartree-exchange-correlation (Hxc) kernel f Hxc ðr 1 ; r 2 ; xÞ. As was established recently [8], a similar efficient description of localized excitations can be achieved in TD Hartree-Fock (TDHF) [9–15] and density matrix functional (TDDMFT) [16–19] theories by the transformation of the HF and natural (NO) orbitals, respectively, to the natural excitation orbitals (NEOs) ha . The latter orbitals are obtained with the diagonalization of the submatrix of the (inverse) response matrix of the TDHF/TDDMFT eigenvalue problem for electronic excitations. This diagonalization is performed in the subspace of the virtual HF orbitals or weakly occupied NOs and the resultant NEOs resemble the virtual KS orbitals. The physical reason behind the established trend is that both virtual KS
orbitals and NEOs represent an electron in the field of N electrons plus an effective hole of the xc origin surrounding the electron, ðN 1Þ electrons in total. For localized excitations this hole efficiently simulates the hole in an excited state left behind by the excited electron. Previously, model effective potentials of ðN 1Þ electrons were employed within the improved virtual orbital (IVO) [20–24] procedure to improve the description of excitations with singly excited electronic configurations. Unfortunately, because of the very same physical reason, the virtual KS orbitals are not suitable for the description of the long-range CT excitations D : A ! Dþ A in the D A complexes [1,25–28]. At the large D A separation R the true CT excitation energy xCT turns to the difference between the vertical ionization potential (VIP) ID of D and the electron affinity A A of A corrected with the electron-hole interaction 1=R
xCT ID A A 1=R:
ð2Þ
The difference ðID A A Þ, which dominates at large R, is exactly rep-
D resented in the rigorous KS theory, ID A A ¼ A S H , with that A D between the energies S and H of the singly occupied molecular
orbital (SOMO) /A S of the anionic A fragment and highest occupied
MO (HOMO) /DH of the D fragment. The former energy is substantially higher than that LA of the lowest unoccupied MO (LUMO) of
A the neutral A, the corresponding positive difference D A ¼ A S L is often called the derivative discontinuity [26,29,30]. This holds true since, as was stated above, an electron on the virtual KS orbital
of the neutral A effectively interacts with ðN A 1Þ electrons, while E-mail address: [email protected] http://dx.doi.org/10.1016/j.cplett.2016.11.034 0009-2614/Ó 2016 Elsevier B.V. All rights reserved.
an electron on the KS orbital of the anion A interacts with N A
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O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
electrons. Because of this, the zero-order TDDFT (1) yields too low estimate of xCT [25]
suitable for the description of CT excitations. The transformation is performed by diagonalizing in the subspace f/a g
TDDFTð0Þ xCT xCT ID A A þ DH LA ¼ D A :
hs
ð3Þ
As was established in [27,30], in order to correct xCT the Hxc kernel f Hxc should diverge in the long-range limit. However, f Hxc of the conventional DFT functionals lack such a divergence, so that conventional TDDFT significantly underestimates xCT . In this Letter the long-range CT problem of TDDFT is illustrated with the example of the paradigmatic D A complex He Be. We consider the CT excitation associated with an electron transfer from the 1s HOAO of the He atom to the 2pp LUAO of the Be atom. Fig. 1 presents the plot of the reference xCISDT values calculated at CT various separations R(He Be) with the configuration interaction (CI) method with all single, double, and triple excitations (CISDT) from two valence MOs and with the 1r2 core frozen. The calculations were performed with the GAMESS (US) package of programs [31] in the aug-cc-pVTZ basis (without the f-functions) [32,33]. The excitation energy xCISDT exhibits the minimum at R = 2.0 Å and its CT further increase amounts to 3.11 eV at 8.0 Å. obtained with the standard Full-linear-response TDDFT xBLYP CT combination BLYP of the exchange functional by Becke (B88) [34] and the correlation functional by Lee, Yang, and Parr (LYP) [35,36] as well as with the corresponding Hxc kernel displays a qualitatively wrong trend (see Fig. 1). Being consistently much lower than xCISDT , the energy xBLYP decreases monotonically with CT CT R, so that the TDDFT-BLYP error increases (in absolute value) from 3.41 eV at R = 1.5 Å to 9.23 eV at R = 8.0 Å. Furthermore, full-linearresponse time-dependent generalized KS (TDGKS) approach with the B3LYP [37] and M06 [38] hybrid functionals also produces substantially underestimated xCT with the deficient dependence on R. Both hybrid functionals yield a displaced and too shallow minimum at R = 3.5 Å, with the absolute xCT error increasing from 3.18 eV at R = 2.0 Å to 7.01 eV at R = 8.0 Å for TDDFT-B3LYP and from 2.17 eV at R = 1.5 Å to 6.64 eV at R = 8.0 Å for TDDFT-M06. In this Letter we introduce a set of CT orbitals (CTOs) fna g, which are obtained with a transformation of the ground-state virtual KS orbitals f/a g of a D A complex and which would be TDDFTð0Þ
PCT" " na
" ¼ n" a na
ð4Þ
^s ðrÞ of D A augof the conventional KS operator h ^ mented with the projected CT (PCT) correction hPCT" ðrÞ the matrix
PCT" hs
^PCT" ðrÞ ¼ h ^s ðrÞ þ h ^PCT" ðrÞ ¼ hðrÞ ^ ^PCT" ðrÞ: h þ v Hxc ð½q; rÞ þ h s
ð5Þ
^ In (5) hðrÞ is the one-electron operator and v Hxc ð½q; rÞ is the Hxc potential of the ground-state density of D A, which consists of ^PCT" reads the Hartree v H and xc v xc parts. The PCT correction h
^PCT" ðrÞ ¼ P b A ½v " ð½qCT ; rÞ v Hxc ð½q; rÞ P bA: h Hxc
ð6Þ
Here, v q ; rÞ is the Hxc spin-up potential of the spinunrestricted KS approach calculated with the density qCT ðrÞ " Hxc ð½
CT
qCT ðrÞ ¼ qðrÞ
X X ci j/"iðDÞ ðrÞj2 þ ca j/"aðAÞ ðrÞj2 ; i2CT
ð7Þ
a2CT
which (with the properly chosen coefficients ci and ca ) represents a combined CT from the occupied (up-spin-) orbitals /"iðDÞ ðrÞ of D A associated with D to the virtual orbitals /"aðAÞ ðrÞ associated with A. It is desirable to apply the PCT correction only to the virtual orbitals localized on A, which participate in the CT D : A ! Dþ A . This bA is achieved by employing in (6) the projection operator P
bA ¼ P
X
j/pA ih/pA j
ð8Þ
bound
onto the bound KS orbitals /pA of the individual fragment A. We would like to stress the following simplifications and virtues of the proposed CTO scheme (4)–(8) compared to the DSCFDFT scheme [39,40], in which the initial qCT of (7) is optimized with the full self-consistent field (SCF) DFT calculation: (1) The transformation of the KS orbitals f/a g to the CTOs fna g amounts to only one cycle of the SCF procedure with a single diagonalization (4).
Fig. 1. The CT excitation energy as a function of R(He Be) calculated with CISDT, CTO, and TDDFT.
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O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
(2) More importantly, the CTO scheme is free from the convergence problems of DSCF-DFT associated with a possible variational collapse to a lower excited configuration as well as to the ground state [41]. During the CTO construction, qCT is not updated. Also, the resultant CTOs are composed from the virtual KS orbitals, so they are orthogonal to the occupied KS orbitals /i , while the latter remain intact. All this makes the abovementioned posibility of the collapse irrelevant, so there is no need in the imposition of constraints, which enforce orbital transitions in the constricted variational DFT (CV-DFT) [42]. The outlined stability of the CTO scheme allows, in principle, to broadly vary the form and composition of qCT to obtain more suitable CTOs in analogy to a variation of the modified one-electron HF operator in the abovementioned IVO procedure [20–24]. Compared to TDHF, the advantage of the CTO scheme is the inclusion of the effect of electron correlation on the calculated orbital energies through the use of some approximate correlation potential within v xc . (3) Last, but not the least, the CTO scheme exploits the full potentiality of one-electron theory. While in DSCF-DFT the excitation energy xCT is calculated as the difference between qCT and the total energies corresponding to DSCFDFT CT q; xCT ¼ E½q E½q, in the CTO scheme xCT is estimated simply as the difference between the orbital energies " n" a and i of the corresponding CTO na and the occupied KS orbital /i n" xCTO a ¼ a i :
ð9Þ
Of course, the CTO estimate (9) requires to use an xc potential v xc , which would yield a good quality orbital energies [43] (see also the presentation below). In its turn, the shape of the CTO represents the density distribution of the excited electron. Could the proposed CTO scheme provide an adequate oneelectron description of the abovementioned paradigmatic CT in the He Be complex? To address this question, the CTOs n"a and the CTO energies n" a are obtained at various R(He Be) separations. The He Be ground state is calculated with the following approximate xc potential
v xcBVWNðSPPÞ
3 2
B VWNðSPPÞ v xcBVWNðSPPÞ ðrÞ ¼ v LDA ðrÞ: x ðrÞ þ 2x ðrÞ þ v c
ð10Þ
where v LDA is the exchange potential of the local density approxix mation (LDA), Bx ðrÞ is the B88 exchange energy density [34], and
v VWNðSPPÞ is the LDA correlation potential of Vosko, Wilk, and Nusair c
(VWN) [44] with the correction of Stoll, Pavlidou, and Preuss (SPP) [45]. The first two terms in (10) approximate the Slater’s exchange potential which, as was shown in [43], yields (when augmented with a suitable correlation potential) good quality orbital energies. The potential (10) is chosen, since its 2s HOAO energy of the Be atom
BVWNðSPPÞ 2s ¼ 9:37 eV perfectly reproduces (as is required in
the KS theory [46–49]) the (minus) first VIP of Be, IBe 1 ¼ 9:32 eV. Fur-
thermore, its 1s energy of the He atom 1s
BVWNðSPPÞ
¼ 25:15 eV fairly
reproduces the (minus) first VIP of He, IHe 1 ¼ 24:59 eV with the rel-
atively small error D1s
BVWNðSPPÞ
¼ 0:56 eV. The PCT correction (6)
of the spinis obtained with the spin-up component v xc unrestricted version of the potential (10) calculated with the following qCT BVWNðSPPÞ"
qCT ðrÞ ¼ qðrÞ j/"2r ðrÞj2 þ 0:5j/"1px ðrÞj2 þ 0:5j/"1py ðrÞj2 :
ð11Þ
The CTO calculations are performed with the original program [50] adapted to GAMESS-US. The data in Fig. 1 and in Tables 1 and 2 gives an affirmative answer to the question posed in the beginning of the previous paragraph. Fig. 1 displays the CTO estimate xCTO CT of xCT BVWNðSPPÞ n" xCTO : CT ¼ 1p 2r þ D1s
ð12Þ
" Here, n" 1p is the energy of the lowest CTO n1p of the p-symmetry localized on Be and 2r is the B-VWN(SPP) energy of the groundstate occupied KS MO /2r associated with the HOAO of He. In (12) it is corrected with the abovementioned B-VWN(SPP) deviation BVWNðSPPÞ D1s . One can see from Fig. 1, that the CTO estimate (12) greatly reduces the large error of conventional TDDFT. Remarkably, CISDT xCTO in the whole range of R CT reproduces well the reference xCT (He Be). Being consistently higher (in average, by 0.77 eV), xCTO CT ðRÞ reproduces very well the dependence of the reference xCISDT on R, with the error of the difference between xCTO CT CT ð8:0Þ and xCTO ð2:0Þ being of only 0.27 eV. CT Based on the CTO scheme (4)–(9), we propose two orbital indices to assess the efficiency of the CT description with the relevant virtual orbital wa of a given basis. The charge-transfer orbital is given with the difference energy index (CTOEI) ICTE a
ICTE ¼ wa n" a a
ð13Þ
between the orbital energies n"a .
wa
of wa and
n" a
of the corresponding
The smaller is the absolute CTOEI value jICTE CTO a j, the better tends to be the description of the CT energy effect with the zeroorder (TD) DFT estimate (1) with wa . The sign of the CTOEI indicates whether (1) tends to overestimate or underestimate xCT . In its turn, the charge-transfer orbital shape index (CTOSI) ICTS is evaluated a CTOðaÞ
with the coefficient caðwÞ
of wa in the expansion of n"a in the consid-
ered basis fwp g CTOðaÞ
ICTS ¼ jcaðwÞ j2 : a
ð14Þ
The larger the CTOSI value is, the better the shape of wa tends to describe the distribution of the excited electron. Tables 1 and 2 compare the variation with R(He Be) of the CTOEI and CTOSI evaluated for the relevant (for the considered CT) B-VWN(SPP) KS and HF 1p-orbitals. According to both CTOEI and CTOSI, the description of the CT with the KS orbital /1p gradually deteriorates with R(He Be) (See Table 1). Indeed, at shorter distance R = 1.5 Å the CTOEI is close to zero, while the CTOSI practically reaches the maximal unit value. The former monotonically increases in absolute value, while the latter decreases with R (He Be), so at R = 8.0 Å the absolute CTOEI value amounts to 4.88 eV, while the CTOSI drops to 0.424. Table 2 displays the opposite trend for the HF orbital v1p . According to both CTOEI and CTOSI, the description of the CT with v1p gradually improves with R(He Be). Indeed, at R = 1.5 Å the STOEI amounts to 6.18 eV, while the CTOSI is only 0.174. The former monotonically decreases, while the latter increases with R (He Be), so at R = 8.0 Å the CTOEI drops to 1.20 eV, while the CTOSI reaches 0.848. The established variation of the proposed CTOEI and CTOSI with R(He Be) makes a qualitative physical sense. Indeed, at shorter R the complex He Be is more similar to compact molecules for which, as was mentioned in the beginning of this Letter, the KS orbitals provide a fair description of excitations. Unlike to this, in the case of long-range CT the excited electron at the affinity level of A is better represented, by virtue of the Koopmans’ theorem, with a HF MO [51,52]. Summarizing, we have proposed a transformation of the virtual KS orbitals to an orbital set (the CTOs) adapted to description of CT excitations. CTOs and their energies are obtained with a single
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O.V. Gritsenko / Chemical Physics Letters 667 (2017) 51–54
Table 1 Variation of the charge-transfer orbital energy (CTOEI, in eV) and shape (CTOSI) indices for the B-VWN(SPP) KS virtual orbital /1p with the He Be separation (in Å). CT orbital indexes/R(He Be)
1.5
2.0
3.5
5.0
1p
5.80
4.36
2.36
1.50
1.08
0.81
5.97 0.17 1.000
5.72 1.36 0.985
5.72 3.36 0.792
5.69 4.19 0.598
5.69 4.61 0.473
5.69 4.88 0.424
n" 1p
CTOEI CTOSI
Table 2 Variation of the charge-transfer orbital energy (CTOEI, in eV) and shape (CTOSI) indices for the HF virtual orbital CT orbital indexes/R(He Be)
n" 1p HF 1p
CTOEI CTOSI
6.5
8.0
v1p with the He Be separation (in Å).
1.5
2.0
3.5
5.0
6.5
8.0
5.80
4.36
2.36
1.50
1.08
0.81
0.38
0.40
0.40
0.39
0.39
0.39
6.18 0.174
4.76 0.214
2.76 0.530
1.89 0.713
1.47 0.813
1.20 0.848
diagonalization of the submatrix of the modified KS operator. The CTO scheme offers a simple estimate of the CT excitation energy with an orbital energy difference. This estimate reproduces well the reference CISDT values in a wide range of the donor-acceptor separations in the paradigmatic He Be complex. At larger R this improvement over the conventional TDDFT is due to the proposed PCT correction, while at the short R it is due to the use of the sufficiently close B-VWN(SPP) approximation to the true KS potential. The CTO-based indices CTOEI and CTOSI have been proposed to assess the suitability of the CT description with virtual orbitals of a given basis set. Both indexes yield the correct trends for the KS and HF orbitals relevant to the considered CT in the He Be. Evert Jan Baerends and Robert van Meer are gratefully acknowledged for critical and stimulating comments. Robert van Meer is also acknowledged for his guidance and assistance in producing and processing of computational data.
[13] [14] [15] [16] [17]
References
[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]
[1] E.J. Baerends, O.V. Gritsenko, R. van Meer, Phys. Chem. Chem. Phys. 15 (2013) 16408. [2] R. van Meer, O.V. Gritsenko, E.J. Baerends, J. Chem. Theory Comput. 10 (2014) 4432. [3] E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52 (1984) 997. [4] M.E. Casida, in: D.P. Chong (Ed.), Recent Advances in Density-Functional Methods, Part I, World Scientific, Singapore, 1995, p. 155. [5] K. Burke, J. Werschnik, E.K.U. Gross, J. Chem. Phys. 123 (2005) 062206. [6] S. Hirata, M. Head-Gordon, Chem. Phys. Lett. 314 (1999) 291. [7] M.A.L. Marques, N.T. Maitra, F.M.S. Nogueira, E.K.U. Gross, A. Rubio, in: Fundamentals of Time-Dependent Density Functional Theory, Lecture Notes in Physics, vol. 837, Springer, Heidelberg, 2012. [8] R. van Meer, O.V. Gritsenko, E.J. Baerends, Chem. Phys. Lett. 639 (2015) 315. [9] J. Frenkel, Wave Mechanics, Advanced General Theory, Clarendon Press, Oxford, 1934. [10] J.P. Blaizot, G. Ripka, Quantum Theory of Finite Systems, MIT Press, Cambridge, Massachusetts, 1986. [11] D.J. Thouless, The Quantum Mechanics of Many-Body Systems, Academic Press, New York, 1972. [12] A. Dalgarno, G.A. Victor, Proc. R. Soc. Lond. A 291 (1966) 291.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
[51] [52]
A. Kerman, S. Koonin, Ann. Phys. 100 (1976) 332. H.B. Schlegel, S.M. Smith, X. Li, J. Chem. Phys. 126 (2007) 244110. H. Eshuis, G. Balint-Kurti, F. Manby, J. Chem. Phys. 128 (2008) 114113. K. Pernal, O. Gritsenko, E.J. Baerends, Phys. Rev. A 75 (2007) 012506. K. Pernal, K. Giesbertz, O. Gritsenko, E.J. Baerends, J. Chem. Phys. 127 (2007) 214101. K.J.H. Giesbertz, O.V. Gritsenko, E.J. Baerends, Phys. Rev. Lett. 105 (2010) 1. K.J.H. Giesbertz, O. Gritsenko, E.J. Baerends, J. Chem. Phys. 136 (2012) 094104. S. Huzinaga, C. Arnau, Phys. Rev. A 1 (1970) 1285. W.J. Hunt, W.A. Goddard III, Chem. Phys. Lett. 3 (1969) 414. K. Morokuma, S. Iwata, Chem. Phys. Lett. 16 (1972) 192. B.J. Mogensen, S. Rettrup, Int. J. Quant. Chem. 44 (1992) 1045. M.W. Schmidt, E. Hull, T. Windhus, J. Phys. Chem. A 119 (2015) 10408. A. Dreuw, J.L. Weisman, M. Head-Gordon, J. Chem. Phys. 119 (2003) 2943. D.J. Tozer, J. Chem. Phys. 119 (2003) 12697. O. Gritsenko, E.J. Baerends, J. Chem. Phys. 121 (2004) 655. N.T. Maitra, J. Chem. Phys. 144 (2016) 220901. D. Vieiera, K. Capelle, C.A. Ullrich, Phys. Chem. Chem. Phys. 11 (2009) 4647. M. Hellgren, E.K.U. Gross, Phys. Rev. A 85 (2012) 022514. M.W. Schmidt, K.K. Baldridge, J. Boatz, S. Elbert, M. Gordon, J. Jensen, S. Kosekt, N. Matsunaga, K. Nguyen, S. Su, T. Windus, M. Dupuis, J. Montgomery, J. Comput. Chem. 14 (1993) 1347. J.T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. D.E. Woon, T.H. Dunning, J. Chem. Phys. 100 (1994) 2975. A.D. Becke, Phys. Rev. A 38 (1988) 3098. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. B. Miehlich, A. Savin, H. Stoll, H. Preuss, Chem. Phys. Lett. 157 (1989) 200. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. Y. Zhao, D.G. Truhlar, Theor. Chem. Acc. 120 (2008) 215. J.C. Slater, Adv. Quant. Chem. 6 (1972) 1. T. Ziegler, A. Rauk, E.J. Baerends, Theor. Chim. Acta 43 (1977) 261. Y.C. Park, M. Krykunov, T. Ziegler, Mol. Phys. 113 (2015) 1636. T. Ziegler, M. Krykunov, I. Seidu, Y.C. Park, Top. Curr. Chem. 368 (2016) 61. O.V. Gritsenko, L.M. Mentel, E.J. Baerends, J. Chem. Phys. 144 (2016) 204114. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. H. Stoll, M. Pavlidou, H. Preuss, Theor. Chem. Acc. 149 (1978) 143. J.P. Perdew, R.G. Parr, M. Levy, J. Balduz, Phys. Rev. Lett. 49 (1982) 1691. J.P. Perdew, M. Levy, Phys. Rev. Lett. 51 (1983) 1884. M. Levy, J.P. Perdew, V. Sahni, Phys. Rev. A 30 (1984) 2745. C.-O. Almbladh, A.C. Pedroza, Phys. Rev. A 29 (1984) 2322. M. Buijse, Electron Correlation (Ph.D. thesis), Vrije Universiteit, Amsterdam, 1991. T. Tsuneda, J.W. Song, S. Suzuki, K. Hirao, J. Chem. Phys. 133 (2010) 174101. T. Stein, H. Eisenberg, L. Kronik, R. Baer, Phys. Rev. Lett. 105 (2010) 266802.