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Natural transition orbitals for the calculation of correlation and excitation energies
Chemical Physics Letters 679 (2017) 52–59
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Research paper
Natural transition orbitals for the calculation of correlation and excitation energies Sebastian Höfener ⇑, Wim Klopper Institute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), P.O. Box 6980, D-76049 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 16 February 2017 In final form 24 April 2017 Available online 26 April 2017
a b s t r a c t We propose a scheme to reduce the virtual space in the second-order approximate coupled-cluster singles-and-doubles (CC2) method, allowing for an accurate description of both ground-state correlation and excitation energies. A set of natural virtual orbitals is constructed using the ground-state MP2 density as well as the configuration-interaction-singles (CIS) excitation vectors. The results show that approximately half of the virtual space can be removed while an accuracy of about 90% for conventional correlation energies is obtained. CC2 excitation energies show in most cases an error of about 2% or less when the virtual space is reduced to half. Ó 2017 Published by Elsevier B.V.
1. Introduction Due to the scaling of correlation methods with basis-set size, many attempts have been made to truncate the size of the virtual orbital space not only for ground-state methods but also for response-property calculations. While first ideas concerning natural orbitals reach back to the 1950s [1–6], in 1997 it was shown by Klopper et al. that natural orbitals based on the second-order Møller-Plesset (MP2) density can provide >90% of the groundstate coupled-cluster singles and doubles (CCSD) correlation energy when using the occupation numbers of the natural virtual orbitals to remove half of the virtual space [7]. Some years later excitation vectors obtained from the configuration-interaction-sin gles (CIS) method were used to set up domains for coupled-cluster methods using localized orbitals [8,9]. Pair-natural orbitals (PNOs) [10–14] and orbital-specific virtual (OSV) orbitals have been (re) discovered and applied by many authors more recently to coupled-cluster methods [15–20] as well as to the random-phase approximation [21]. These methods are able to significantly reduce the virtual space and thus yield dramatically reduced computation times. However, a drawback of several methods is that the efficiency comes at the cost of non-orthonormal orbital sets for different orbital pairs and excited states, requiring special implementations. To circumvent this, a restricted-virtual space (RVS) approach was suggested by Send et al. who reduced the virtual space drastically using only the canonical virtual orbitals with orbital energies below 50 eV [22]. This method provides accurate ⇑ Corresponding author. E-mail addresses: [email protected] (S. Höfener), [email protected] (W. Klopper). http://dx.doi.org/10.1016/j.cplett.2017.04.083 0009-2614/Ó 2017 Published by Elsevier B.V.
excitation energies in many cases but ground-state correlation energies as well as frequency-dependent dipole polarizabilities using coupled-cluster theory can exhibit large errors [23] and it can lead to increased errors for np or pr states using ADC(2) [24]. In the present work, we construct effective natural transition orbitals (NTOs) suited for both ground and excited states via combining the ground-state MP2 density with CIS excitation vectors and a subsequent diagonalization of the virtual-virtual block of the so-obtained Fock matrix to form canonical orbitals. To the best of our knowledge, such a scheme is applied for the first time to calculate G0W0 quasi-particle energies. As an excited state property, we investigate oscillator strengths obtained at the second-order approximate coupled-cluster singles-and-doubles (CC2) level of theory [25]. 2. Method In order to reduce the space of virtuals efficiently, the key step is to combine appropriate measures for ground-state correlation energies and excitation energies, from which those virtual orbitals are extracted which belong to occupation numbers above a certain threshold. In the following, we discuss three methods to construct effective virtual spaces suited for both ground and excited states with a subsequent diagonalization of the so-obtained Fock matrix to form canonical orbitals. 2.1. Ground-state density For coupled-cluster methods using single and double excitations, it was shown by Klopper et al. that correlation calculations using only the natural virtual orbitals of the unrelaxed virtual-
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
virtual block of the MP2 ground-state density matrix (abbreviated ‘‘MP2 density matrix” throughout the present work),
DMP2 ¼ ab
X
DMP2;kern ; ij;abc
ð1Þ
ijc
with large occupation numbers yield a large fraction of the correlation energy [7]. The kernel is given as
DMP2;kern ¼ ij;abc
2ðiajg 12 jjcÞ ðicjg 12 jjaÞ 2ðibjg 12 jjcÞ : ac ij bc ij
ð2Þ
Here, g 12 denotes the Coulomb operator and pq is the sum of orbital energies of orbitals p and q, respectively. In order to reduce the scaling, it has also been proposed to use the diagonal contributions only [16],
DdMP2 ¼ ab
X MP2;kern Dii;abc ;
ð3Þ
53
and from the full set those vectors are collected that belong to singular values above a given threshold, while singular values belonging to a singular value below the threshold are removed. Note that an upper bound of the rank of the matrix RðnÞ is given by minðnvir ; noct Þ, that is, the minimum value of the number of virtual and (active) occupied orbitals. To give an example, for the helium atom only one particle NTO can be obtained, independent of the size of the basis. These excited-state NTOs are combined with a set of NVOs based on the MP2 density matrix,
X ¼ XMP2 Vð1Þ Vð2Þ . . . Vðnexc Þ :
ð9Þ
~ is calculated and linear For this set of orbitals the overlap S dependencies are removed using SVD,
~¼U ~R ~T; ~V XT X ¼ S
ð10Þ
~ which belong to singular values by deleting those vectors from V
ic
but, after observing a significant deterioration in accuracy, the full MP2 density matrix is used throughout the present work.
smaller than a fixed threshold of 108 to avoid numerical insta~ ~ together with an bilities, which yields the rectangular matrix V
2.2. Excited-state density
~ A linear-independent set of ~ appropriate eigenvalue matrix R. vectors is then obtained as
In the following, we discuss three main approaches, where for all schemes it is assumed that the CIS excitation vectors ð1Þ
~~ ~~ 1=2 Y ¼ XV R :
ð11Þ
ðnexc Þ
R ...R for the lowest nexc excitations in the full virtual space are available if needed. 1. (Augmented) RVS: Use the RVS approximation as proposed by Send et al. [22] and include additionally the natural virtual orbitals (NVOs) obtained from a diagonalization of the full MP2 density matrix,
XTMP2 DMP2 XMP2 ¼ d;
ð4Þ
where d is a diagonal matrix containing the eigenvalues and XMP2 are the eigenvectors. From the full set (XMP2 ) those eigenvectors (YMP2 ) are collected that belong to eigenvalues above a given threshold, while eigenvectors belonging to an eigenvalue below the threshold are removed. Finally, the two sets are joined and linear dependencies are removed. 2. Natural effective-density based orbitals (NEO): Using the ground-state MP2 density and an effective excited-state(s) density,
Dex ab ¼
nexc X X ðnÞ ðnÞ Rai Rbi ; n¼1
Finally, the orbitals obtained from the three schemes are rotated such that they diagonalize the virtual-virtual block of the Fock matrix, so that they can be used in a standard coupledcluster implementation assuming canonical orbitals provided that virtual orbitals can be frozen, i.e., omitted. 3. Results The proposed schemes have been implemented in the KOALA program package [26]. For all methods, the resolution-of-identity (RI) is used. First, the MP2-based NVOs are applied to the calculation of different ground-state correlation schemes to illustrate their broad applicability and to validate our implementation. Second, CC2 excitation energies and oscillator strengths are investigated using the new orbitals. All quantum-chemical methods used in the following are well-known in the literature, in the present work only the performance of the natural orbitals as proposed is assessed.
ð5Þ
i
3.1. Ground-state correlation energies
an effective density matrix is constructed using appropriate weights:
Deff ¼ wðMP2Þ DMP2 þ wðexÞ Dex : In the present work, the weights w
ð6Þ ðMP2Þ
ðexÞ
and w
were empiri-
cally chosen to be 1. The effective density matrix Deff is then diagonalized,
XT Deff X ¼ d;
ð7Þ
where d is a diagonal matrix containing the eigenvalues and X are the eigenvectors. From the full set (X) those eigenvectors (Y) are collected that belong to eigenvalues above a given threshold, while eigenvectors belonging to an eigenvalue below the threshold are removed. 3. Natural transition orbitals (NTOs): A singular value decomposition (SVD) is applied to each of the lowest nexc CIS excitation vectors, T
RðnÞ ¼ UðnÞ RðnÞ VðnÞ ;
ð8Þ
We begin with investigating the performance of NVOs for frozen-core RI-MP2 and RI-CCSD correlation energies calculated for the water molecule using the cc-pVXZ-F12 (X = D,T,Q) basis sets [27], see Table 1. The table shows that already small subsets can yield large fractions of the correlation energy. For example, for the cc-pVQZ-F12 basis a threshold of 1 104 yields 40 natural virtual orbitals, which corresponds to 27%, providing 94% (95%) of the MP2 (CCSD) correlation energy, which is similar to earlier work [7]. A striking efficiency of the MP2 NVOs is observed when the explicitly-correlated RI-MP2-F12/3B-sp method [28] is used. For MP2 the conventional convergence with respect to the basis is observed while for MP2-F12/3B-sp quasi-converged results are obtained for the double-zeta basis using the MP2 NVOs. For example, the result obtained in the cc-pVDZ-F12 basis employing NVOs with a threshold of 5 104 (17 virtual orbitals) corresponds to > 96% of the best value computed in the cc-pVQZ-F12 basis without truncation of the virtual space (150 virtual orbitals). This is rooted in the large redundancy of the conventional and the explicitly-correlated double excitations [29]. One comment on
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S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 1 Frozen-core correlation energies in mEh of the water molecule (d(O–H) = 1.8168 Bohr, a(HOH) = 104.109°) as calculated using the KOALA program using different basis sets. Values in parentheses are the number of virtual orbitals. Approx.e
Method RI-MP2
MP2 NVO
Thr.x 4
1 104
5 10
RI-MP-2F12
MP2 NVO
MP2 NVO
b c d e x
y
224.4 (17)
239.9 (20)
243.8 (21)
236.5 (25)
262.7 (33)
271.1 (40)
241.5 (43)
273.3 (84)
287.2 (150)
290.4 (17)
292.2 (20)
293.1 (21)
1 104
293.4 (25)
296.6 (33)
297.4 (40)
–
296.1 (43)
299.5 (84)
300.5 (150)
5 104
230.2 (17)
246.6 (20)
250.7 (21)
1 104
242.3 (25)
269.0 (33)
277.1 (40)
–
246.7 (43)
277.6 (84)
289.4 (150)
RI-CCSD a
cc-pVQZ-F12c
–
d
RI-CCSD
cc-pVTZ-F12b
5 104
RI-MP2 RI-MP2-F12d
cc-pVDZ-F12a
Geminal exponent c ¼ 0:9. Geminal exponent c ¼ 1:0. Geminal exponent c ¼ 1:1. RI-MP2-F12/2B-sp method, see Ref. [28]. Approximation for the virtual space. Threshold for removing virtual orbitals. Ref. [46].
the projector used in F12 theory is necessary at this point. When the virtual space is truncated, virtual orbitals might be removed from the orbital set that are not covered by the complementary auxiliary basis set (CABS). In order to avoid ‘‘holes” in the unified set of orbital basis and CABS, all basis functions belonging to the orbital basis are included in the CABS space. In case of a full virtual space the redundancy is eliminated automatically by the SVD orthogonalization during the CABS construction [30] while for truncated virtual spaces the removed virtual space is included in the CABS. A method which has experienced some attention in the past is given by the random-phase approximation (RPA), which can be obtained in a sum-over-states manner when all excited states are computed [31–34]. The results for correlation energies calculated using RPA are collected in Table 2, employing Eqs. (27) and (28) of Ref. [34],
X 1 X 0 xTDHF x0CIS ð12Þ 2 n o X X X X 1 ¼ x0TDHF x0CIS þ n x1TDHF x1CIS ; ð13Þ 4
EdRPA ¼ c ErCCD c
P 0 P where x and x1 denote the sums of the singlet and triplet excitation energies of the method given in the subscript, respecflip flip while n ¼ 3 yields ErCCD;w [34]. Note tively; n ¼ 1 yields ErCCD;w=o c c that the spin-flipped excitations do not contribute to the dRPA cor-
relation energy. Although the absolute values of the sums of the excitation energies are changed by hundreds of hartree when NVOs are used, the final correlation energy shows an accuracy of about 85% when only 38% of the virtual orbitals are included. When (vertical) ionization energies are to be studied, the manybody Green’s function GW formalism has been shown to provide reliable results, which has been applied to molecules recently [35,36]. In this method, quasi-particle (QP) energies are calculated as Kohn-Sham (KS) orbital energies shifted using, e.g., TDA or RPA excitation vectors according to KS KS KS QP p ¼ p þ Z p h/p jRX þ RC ðp Þ V xc j/p i:
ð14Þ
Note that using NVOs, the truncated virtual space is used for the excited states only, while all other terms are computed using the original, that is, untruncated Kohn-Sham MOs, see appendix. Results for this method are collected in Table 3. To give an example, for the rpa-G0W0 method 14 NVOs yield a quasi-particle energy QP ðr1s Þ of 403:5 eV, while the final result using 55 virtual orbitals is 402:2 eV, exhibiting an error of 1:2 eV. Tightening the threshold to 1 104 decreases the error to below 0:3 eV. For the highest occupied molecular orbital (HOMO), QP ðr2p Þ, a reduced dependence on the virtual space is observed using NVOs, for which the biggest error is below 0:05 eV for rpa-G0W0, while for the RVS method the error amounts about 0:3 eV. A different behavior is
Table 2 All-electron RI-RPA and RI-dRPA correlation energies in Eh of the N2 molecule (d(N–N) = 2.0749 bohr), cf. Ref. [34]. All calculations employ a def2-TZVPP basis with corresponding auxiliary basis sets. All energies are based on a Hartree-Fock determinant and MP2 natural orbitals as obtained from the full virtual-virtual block of the MP2 density. Method Thr.b nvir a P 0
xTDHF x1 P TDHF x0 P CIS x1CIS P
Without flipc With flipd a b c d
RI-dRPA 4
5 10 21 (147) 1026.20
4
1 10 46 (322) 3017.49
RI-RPA
55 (385) 3555.64
5 10 21 (147) 942.53
1 104 46 (322) 2833.28
55 (385) 3346.33 3327.41
Full
4
Full
1016.95
3000.06
3537.50
932.71
2815.08
1026.90
3018.41
3556.57
943.18
2834.02
3347.08
1016.95
3000.06
3537.50
933.23
2815.68
3328.02
0.35
0.46
0.47
0.29 0.55
0.34 0.64
0.34 0.65
Number of virtual orbitals; parentheses: number of excitation vectors. Threshold for removing virtual orbitals. Without spin-flipped (de) excitations. With spin-flipped (de) excitations.
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Table 3 All-electron PBE G0W0 quasi-particle energies QP in eV of the N2 molecule (d(N–N) = 2.0749 bohr), cf. Ref. [34]. All calculations employ a def2-TZVPP basis with corresponding auxiliary basis sets and grid 5 for the DFT integration. MP2 natural orbitals as obtained from the full virtual-virtual block of the MP2 density. The damping parameter d was set to zero in all calculations. Method Thr.b nvir a
QP ðr2p Þ QP ðp2p Þ QP ðr2p Þ QP ðp2p Þ QP ðr2s Þ QP ðr2s Þ QP ðr1s Þ QP ðr1s Þ a b c
tda-G0W0
rpa-G0W0
1 103 14 (98)
5 104 17 (119)
1 104 46 (322)
RVSc
Full
5 104 17 (119)
1 104 46 (322)
RVSc
Full
55 (385)
1 103 14 (98)
22 (154)
22 (154)
55 (385)
8.7
8.7
8.6
3.4
3.2
2.8
8.5
8.4
9.1
9.0
8.9
8.9
8.8
3.1
2.6
3.5
3.4
3.0
3.2
14.0
14.1
2.8
14.0
13.7
14.0
14.8
14.9
14.8
14.5
14.8
16.8 16.8
16.9
16.7
16.3
16.6
16.4
16.4
16.3
15.9
16.3
16.8
16.7
16.3
16.6
17.9
17.9
17.8
17.5
17.8
29.4 397.4
29.4 397.2
30.0 396.1
29.8 398.2
29.9 395.8
30.7 403.4
30.7 403.3
30.8 402.4
29.0 403.9
29.0 402.1
397.4
397.3
396.2
398.2
395.9
403.5
403.4
402.5
403.9
402.2
Number of virtual orbitals; parentheses: number of excitation vectors. Threshold for removing virtual orbitals. Threshold for orbital energies: 50 eV.
observed for the lowest unoccupied orbital (LUMO),
QP p2p
,
however. Here, NVOs lead to an error slightly above 0:2 eV employing a threshold of 1 104 , while RVS increases the error to about 0:5 eV and 0:4 eV for tda-G0W0 and rpa-G0W0, respectively. The table also reveals that no systematic improvement is obtained when additional natural orbitals are used, for example, in case of tda-G0W0 a threshold of 1 103 yields an error of 0:02 eV for the HOMO, while a threshold of 5 104 results in an error of about 0:1 eV. However, the errors introduced using truncated virtual spaces using a threshold of 5 104 remain for the occupied orbitals significantly below the method error in case of tda-G0W0 when the rpa-G0W0 values are taken as reference. For example, the truncated virtual space introduces an error of about 0:1 eV for the HOMO for tda-G0W0, while the difference of full-space tda-G0W0 and rpa-G0W0 is as large as 0:8 eV. In our test system, the only exception is the quasi-particle energy of the LUMO which exhibits an error of about 0:6 eV for a threshold of 5 104 compared to full-space tda-G0W0, while the difference of full-space tda-G0W0 and rpa-G0W0 amounts ca. 0:2 eV. 3.2. Natural orbitals for CC2 ground-state correlation and excitation energies Having investigated the performance of NVOs for which only the MP2 density was used, we now turn our attention to excited states, for which also further virtual orbitals are included to provide a balanced description for both ground and excited states. First, results for the benzene and DMABN molecule are collected in Tables 4 and 5, respectively, both calculated using frozen-core RICC2 [25,37] and Dunning’s aug-cc-pVTZ basis [38,39]. Both tables also reveal that both the original RVS and augmented RVS methods (applying a threshold of 50 eV) yield correlation energies which deviate significantly while excitation energies show a sufficient accuracy. Note that the large deviation for correlation energies is expected for RVS. Concerning higher excited states, both tables also reveal that the NEO approach seems to yield the largest errors of about 0.2 eV, but the virtual space is by far the smallest. Furthermore, the (augmented) RVS method seems to introduce small errors for oscillator strengths for DMABN for the CT excitation. Second, the ethylene molecule is investigated with respect to the 11 B3u and 21 B3u excited states exhibiting valence and Rydberg character, respectively. The Rydberg basis ‘‘CM2”, which was added to the standard def2-TZVPPD basis, consists of 2s2p2d with exponents 0:01 and 0:0033 [40], respectively, which are placed in the
center of mass of the molecule [41]. It was shown that the CIS method, which is used in the present work as a prescreening method, is able to capture the Rydberg state of the ethylene molecule if a proper Rydberg basis is included [42]. While the conventional basis yields virtual orbitals with orbital energies of about 0.1 eV and higher, the CM2 basis provides virtual orbitals with orbital energies in the between the eigenvalue of the HOMO and about 0.1 eV. Results for RI-CC2 ground-state correlation energies and excitation energies are collected in Table 6. The table confirms that the original RVS approach seems to yield rather large errors in the oscillator strengths and that the NEO approach yields by far the smallest virtual space. It also illustrates how different the requirements for ground-state correlation energies and (Rydberg) excitations can be: While the ground-state correlation energy shows an error of about 0.1% using a threshold of 1 105 , the Rydberg excitation exhibits an error of more than 4 eV (44%) when only the ground-state MP2 density is used. This large error arises because the diffuse functions are not important for the ground-state correlation energy and are thus not included even for a rather tight threshold of 1 105 , highlighting the need to include measures for individual excited states. Note, however, that the Rydberg excitation is the twelfth excitation and thus at least 12 excitations have to be calculated using C 1 symmetry. Using symmetry, only the lowest two excitations in B3u would be calculated, reducing the virtual space even further. Such an additional reduction was achieved by taking into account the MP2 density and the excitation vector of the Rydberg state only, see Table 6, leading to virtual spaces that are well suited for ground-state correlation energies and the target Rydberg state showing errors of about 5% and 0.1 eV, respectively, while the valence state, which was not taken into account for the determination of the reduced virtual space, exhibits errors of 0.4 eV and larger. Finally, RI-CC2 calculations using the Karlsruhe def2-SVPD and def2-TZVP basis sets [43,44] are carried out for the lowest singlet state (S1 ) of deprotonated retinal in vacuum, of which the geometry displayed in Fig. 1 was taken from Ref. [45]. Data including the original RVS method of Send et al. is collected in Table 7. It confirms that the (augmented) RVS-based method performs well for the excitation energies in almost all cases but can yield large deviations in the correlation energies even for rather tight thresholds, cf. Ref. [23]. To give an example, while RVS using a 103 threshold yields 534 virtual orbitals and a correlation energy of 3.112 Eh corresponding to 84%, NEO using a threshold of 104 yields 523 virtual orbitals and a correlation energy of 3.673 Eh corresponding to 99%. This again illustrates how important the
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S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 4 The first (pp B2u ), fifth (pr A2u ), and tenth (pp E1g ) excitation energies in eV of the benzene molecule (d(C–C) = 2.6399 Bohr, d(C–H) = 2.0485 Bohr) as calculated using the KOALA program employing Dunning’s aug-cc-pVTZ basis. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Method RI-CC2
c d e x y
Thr.x
nvir c
Corr.a
pp B2u b
pr A2u b
pp E1g b
1 5 104
187 246
639.1 855.9
5.32 (0.00) 5.25 (0.00)
6.71 (0.07) 6.94 (0.07)
7.12 (0.63) 7.13 (0.66)
1 104
272
902.7
5.24 (0.00)
6.98 (0.07)
7.13 (0.66)
99
858.9
5.29 (0.00)
6.91 (0.07)
7.33 (0.71)
1 104
174
944.2
5.24 (0.00)
6.99 (0.07)
7.21 (0.70)
5 104
198
892.4
5.24 (0.00)
6.92 (0.07)
7.13 (0.66)
1 104
297
962.3
5.22 (0.00)
6.98 (0.07)
7.13 (0.66)
RI-CC2
–
393
972.9
CCSDy
–
RI-CC2
a
(a) RVS
e
5 104
RI-CC2
b
Approx.d
NEO
NTO
5.22 (0.00)
6.99 (0.06)
7.12 (0.66)
5.14 (0.00)
6.99 (0.07)
7.29 (1.33)
Correlation energy in mEh . Excitation energy in eV. Number of virtual orbitals. Approximation for the virtual space. Threshold for orbital energies: 50 eV. Threshold for removing virtual orbitals. Ref. [46].
Table 5 The first (1 B) and second (1 A) excitation energy in eV of the 4-(N; N-dimethylamino) benzonitrile (DMABN) molecule as calculated using the KOALA program employing Dunning’s aug-cc-pVTZ basis. The geometry was taken from Ref. [47]. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Thr.x
nvir c
Corr.a
1 5 104
321 426
1.219 1.672
4.42 (0.03) 4.41 (0.03)
4.46 (0.15) 4.59 (0.32)
1 104
478
1.762
4.39 (0.03)
4.61 (0.37)
4
5 10
166
1.676
4.48 (0.02)
4.71 (0.40)
1 104
302
1.843
4.42 (0.02)
4.67 (0.46)
5 104
199
1.638
4.44 (0.03)
4.62 (0.30)
1 104
374
1.857
4.41 (0.03)
4.65 (0.44)
RI-CC2
–
697
1.898
4.40 (0.03)
4.64 (0.45)
Exp.y
–
4.25
4.56
Method RI-CC2
RI-CC2
RI-CC2
a b c d e f x y
Approx.d (a) RVS
NEO
NTO
f
1
B (L)b,e
1
A (CT)b,e
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Approximation for the virtual space. Local (L) and charge-transfer (CT) excitation, cf. Ref. [47]. Threshold for orbital energies: 50 eV. Threshold for removing virtual orbitals. Ref. [48].
MP2 density is for a balanced description of ground and excited states. Results for the NEO and NTO approach are collected in Tables 8 and 9, respectively. Both approaches are related to each other and thus yield similar but not identical results. In case of NEOs, 359 virtual orbitals for the def2-SVPD basis (threshold 5 104 ) yield a correlation energy of 3.510 Eh , while 368 NTOs (threshold 1 103 ) yield a correlation energy of 3.391 Eh . On the other hand, for this example NEOs yield an excitation energy of 3.72 eV while NTOs yield 3.68 eV. Both formulations reduce the virtual space to about half of its original size for the cases investigated, while NTOs seem to perform slightly better than NEOs for excitation energies while NEOs seem to perform slightly better than NTOs for correlation energies.
4. Conclusions The investigations presented show that simplistic schemes to truncate the virtual space can lead to results that are biased either
to the ground state or to excited states. A pragmatic solution is to combine information from different estimates and span a subspace to find a compromise, corresponding to virtual orbitals obtained from the MP2 ground-state density and virtual orbitals obtained from CIS excitation vectors. This approach can be formulated in two ways, either via an effective density or via natural transition orbitals with subsequent orthonormalization. Both formulations reduce the virtual space to about half of its original size for the cases investigated, while NTOs seem to perform slightly better than NEOs for excitation energies. On the other hand, NEOs seem to perform slightly better than NTOs for correlation energies. However, the CIS treatment must provide a good zeroth-order description for both methods which is the case for HOMO-LUMO transitions, for instance. Concerning the calculation of quasi-particle energies, the use of NVOs shows a varying performance depending on the molecular orbital under investigation. While for the N2 molecule the errors are 0.1 eV or less for the HOMO, the errors for other orbitals can increase to more than 1 eV using truncated virtual spaces. Although a single test case is not able to validate whether the accu-
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Table 6 11 B3u (valence state) and 21 B3u (Rydberg state) excitation energy in eV of the ethylene (C2H4) molecule employing the def2-TZVPPD + CM2 basis, see main text. The geometry was taken from Ref. [42]. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Method/ Approx.d RI-CC2/MP2
RI-CC2/(a) RVS
Thr.a
nvir b
Corr.c
11 B3u (valence)
21 B3u (Rydberg)
1 103
24
277.6
10.31 (0.437)
19.34 (0.153)
1 104
65
364.5
8.90 (0.424)
15.22 (0.128)
1 105
109
375.1
8.51 (0.414)
13.24 (0.088)
1
80 95
215.9 285.9
7.95 (0.364) 8.01 (0.294)
8.91 (0.046) 9.04 (0.036)
1 103
RI-CC2/NEO
RI-CC2/NTO
RI-CC2/NEO
RI-CC2/NTO
1 104
107
318.7
8.03 (0.377)
9.11 (0.031)
1 105
122
341.4
8.03 (0.379)
9.15 (0.028)
1 103
38
279.0
8.21 (0.381)
9.07 (0.067)
1 104
79
364.0
8.09 (0.387)
9.16 (0.032)
1 105
126
375.2
8.04 (0.382)
9.17 (0.026)
3
101
322.7
8.03 (0.378)
9.11 (0.030)
1 104
135
372.5
8.03 (0.380)
9.17 (0.027)
1 105
152
375.6
8.03 (0.381)
9.17 (0.026)
1 104
66
363.9
8.81 (0.385)
9.18 (0.052)
1 105
114
375.2
8.42 (0.393)
9.18 (0.031)
1 104
73
365.3
8.79 (0.388)
9.17 (0.048)
1 105
117
375.2
8.42 (0.392)
9.18 (0.031)
–
152
375.6
8.03 (0.381)
9.17 (0.026)
1 10
x
x
RI-CC2 a b c d x
Threshold for removing virtual orbitals. Number of virtual orbitals. Correlation energy in mEh . Approximation for the virtual space. Only the MP2 density and the excitation vector for the Rydberg excitation are used.
Fig. 1. Ground-state geometry of the deprotonated rhodopsin as taken from Ref. [45]. Nitrogen atoms are plotted in blue, carbon atoms in gray, and hydrogen atoms in white. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 7 The first excitation energy in eV of the rhodopsin system at RI-CC2/(a) RVS level of theory as calculated using the KOALA program. The threshold for orbital energies is 50 eV. A threshold of 1 corresponds to the original RVS method [22]. Thr.x
def2-SVPD nvir
1
a b c x
c
a
def2-TZVP Corr.a
S1b
340 534
2.691 3.609
3.64 3.62
Corr.
S1b
nvir
c
1 103
456 534
2.647 3.112
3.70 3.67
5 104
574
3.252
3.66
620
3.842
3.61
1 104
647
3.516
3.65
671
4.111
3.61
1 105
695
3.670
3.65
848
4.142
3.60
–
702
3.694
3.65
903
4.178
3.60
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
58
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 8 The first excitation energy in eV of the rhodopsin system at RI-CC2/NEO level of theory as calculated using the KOALA program. Thr.x
def2-SVPD nvir
a b c x
c
def2-TZVP S1b
a
Corr.
Corr.a
S1b
297
3.499
3.72
391
3.757
3.68
671
4.111
3.62
3.65
878
4.176
3.60
3.65
903
4.178
3.60
3
1 10
277
3.296
3.76
5 104
359
3.510
3.72
1 104
523
3.673
3.66
1 105
682
3.694
–
702
3.694
nvir
c
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
Table 9 The first excitation energy in eV of the rhodopsin system at RI-CC2/NTO level of theory as calculated using the KOALA program. Thr.x
def2-SVPD nvir
a b c x
c
def2-TZVP
a
S1b
Corr.
nvir
b
Corr.a
S1b
3
1 10
368
3.391
3.68
388
3.610
3.64
5 104
460
3.576
3.66
495
3.848
3.62
1 104
642
3.688
3.65
793
4.147
3.61
1 105
702
3.694
3.65
903
4.178
3.60
–
702
3.694
3.65
903
4.178
3.60
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
racy obtained using NVOs is sufficient, the numerical example reveals that a simple truncation of the virtual space using RVS can lead to large error bars in particular for the HOMO, from which the ionization energy is to be deduced, and does thus not suffice. We expect a similar performance of NVOs and RVS for advanced GW schemes beyond the G0W0 approximation.
h/KS p j RC
XX j/KS KS p i ¼ p
2 KS j /KS p /k jqn j
n
k
þ
KS KS p k þ xn id
2 KS j /KS p /c jqn j
XX c
n
KS KS p c xn þ id
;
ð18Þ
where n runs over all excited states, and thus
Acknowledgments S.H. gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft DFG (HO-4605/2-1)
D E XX KS /KS j/ ¼ p jð@ RC ðÞ=@ Þ¼KS p p k
Appendix A
n
2 KS j /KS p /k jqn j
KS KS p k þ xn id
XX
2
2 KS jð/KS p /c jqn Þj KS ðKS p c xn þ idÞ
2
c
n
:
A.1. Equations used for G0W0
ð19Þ
In the G0W0 method using natural orbitals, the quasi-particle energies are calculated according to:
The natural orbitals only enter the calculation of the excitedstates transition densities:
KS KS KS QP p ¼ p þ Z p h/p jRX þ RC p
jqn Þ ¼
V xc j/KS p i;
ð15Þ
KS where /KS p and p are orbitals and orbital energies of the original basis, respectively. This leads to the following working equations:
D
E X KS KS KS KS KS /KS /p /i j/i /p ¼ p jRX j/p
ð16Þ
i
n o1 j/KS : Z p ¼ 1 h/KS p jð@ RC ðÞ=@ Þ¼KS p i p The correlation part of the self energy is computed as:
ð17Þ
X NO NO NO 1=2 ðnÞ j/b /j ÞðNO Z bj x1=2 ; b j Þ n
ð20Þ
bj
obtained from the RPA eigenvalue problem in the truncated (NO) space:
M ZðnÞ ¼ ZðnÞ x2n ;
ð21Þ
where
M ¼ ðA BÞ1=2 ðA þ BÞ ðA BÞ1=2 :
ð22Þ
The (closed-shell) orbital-rotation Hessians are given as:
Aai;bj ¼
NO NO a i
Bai;bj ¼ 2ðaNO i
NO
NO
NO NO
dij dab þ 2ðaNO i jb j Þ
NO NO
jb j Þ:
ð23Þ ð24Þ
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
References [1] P.-O. Löwdin, Correlation problem in many-electron quantum mechanics I. review of different approaches and discussion of some current ideas, in: I. Prigogine (Ed.), Advances in Chemical Physics, vol. 2, John Wiley & Sons, Inc., Hoboken, NJ, USA, 1958, http://dx.doi.org/10.1002/9780470143483.ch7. [2] W. Kutzelnigg, Theoret. Chim. Acta 1 (4) (1963) 327–342. [3] C.F. Bender, E.R. Davidson, J. Phys. Chem. 70 (8) (1966) 2675–2685. [4] C.F. Bender, E.R. Davidson, J. Chem. Phys. 47 (12) (1967) 4972–4978. [5] C.F. Bender, E.R. Davidson, Phys. Rev. 183 (1969) 23–30. [6] S.A. Houlden, I.G. Csizmadia, Theoret. Chim. Acta 30 (3) (1973) 209–216. [7] W. Klopper, J. Noga, H. Koch, T. Helgaker, Theor. Chem. Acc. 97 (1997) 164. [8] T. Korona, H.-J. Werner, J. Chem. Phys. 118 (7) (2003) 3006–3019. [9] D. Kats, T. Korona, M. Schütz, J. Chem. Phys. 125 (10) (2006) 104106. [10] W. Meyer, Int. J. Quant. Chem. 5 (S5) (1971) 341–348. [11] W. Meyer, J. Chem. Phys. 58 (3) (1973) 1017–1035. [12] R. Ahlrichs, H. Lischka, V. Staemmler, W. Kutzelnigg, J. Chem. Phys. 62 (4) (1975) 1225–1234. [13] R. Ahlrichs, F. Driessler, H. Lischka, V. Staemmler, W. Kutzelnigg, J. Chem. Phys. 62 (4) (1975) 1235–1247. [14] R. Ahlrichs, F. Keil, H. Lischka, W. Kutzelnigg, V. Staemmler, J. Chem. Phys. 63 (1) (1975) 455–463. [15] F. Neese, F. Wennmohs, A. Hansen, J. Chem. Phys. 130 (11) (2009) 114108. [16] J. Yang, Y. Kurashige, F.R. Manby, G.K.L. Chan, J. Chem. Phys. 134 (4) (2011) 044123. [17] C. Krause, H.-J. Werner, Phys. Chem. Chem. Phys. 14 (2012) 7591–7604. [18] D.P. Tew, C. Hättig, Int. J. Quant. Chem. 113 (3) (2013) 224–229. [19] B. Helmich, C. Hättig, J. Chem. Phys. 135 (21) (2011) 214106. [20] G. Schmitz, B. Helmich, C. Hättig, Mol. Phys. 111 (16–17) (2013) 2463–2476. [21] M. Kállay, J. Chem. Phys. 142 (20) (2015) 204105. [22] R. Send, V.R.I. Kaila, D. Sundholm, J. Chem. Phys. 134 (21) (2011) 214114. [23] A. Kumar, T.D. Crawford, J. Phys. Chem. A 121 (3) (2017) 708–716. [24] C. Yang, A. Dreuw, J. Comput. Chem. (2017), http://dx.doi.org/10.1002/ jcc.24794.
59
[25] O. Christiansen, H. Koch, P. Jørgensen, Chem. Phys. Lett. 243 (5–6) (1995) 409– 418. [26] S. Höfener, J. Comput. Chem. 35 (23) (2014) 1716–1724. [27] K.A. Peterson, T.B. Adler, H.-J. Werner, J. Chem. Phys. 128 (8) (2008) 084102. [28] R.A. Bachorz, F.A. Bischoff, A. Glöß, C. Hättig, S. Höfener, W. Klopper, D.P. Tew, J. Comput. Chem. 32 (11) (2011) 2492–2513. [29] S. Höfener, D.P. Tew, W. Klopper, T. Helgaker, Chem. Phys. 356 (13) (2009) 25–30. [30] E.F. Valeev, Chem. Phys. Lett. 395 (46) (2004) 190–195. [31] Z. Yan, J.P. Perdew, S. Kurth, Phys. Rev. B 61 (2000) 16430–16439. [32] F. Furche, Phys. Rev. B 64 (2001) 195120. [33] F. Furche, J. Chem. Phys. 129 (11) (2008) 114105. [34] W. Klopper, A.M. Teale, S. Coriani, T.B. Pedersen, T. Helgaker, Chem. Phys. Lett. 510 (13) (2011) 147–153. [35] M.J. van Setten, F. Weigend, F. Evers, J. Chem. Theory Comput. 9 (1) (2013) 232–246. [36] K. Krause, M.E. Harding, W. Klopper, Mol. Phys. 113 (13-14) (2015) 1952– 1960. [37] C. Hättig, F. Weigend, J. Chem. Phys. 113 (13) (2000) 5154–5161. [38] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [39] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [40] O. Christiansen, H. Koch, A. Halkier, P. Jorgensen, T. Helgaker, A. Sánchez de Merás, J. Chem. Phys. 105 (16) (1996) 6921. [41] K. Kaufmann, W. Baumeister, M. Jungen, J. Phys. B: At. Mol. Opt. Phys. 22 (14) (1999) 2223–2240. [42] J.S. Kottmann, S. Höfener, F.A. Bischoff, Phys. Chem. Chem. Phys. 17 (2015) 31453–31462. [43] F. Weigend, M. Häser, H. Patzelt, R. Ahlrichs, Chem. Phys. Lett. 294 (1998) 143. [44] D. Rappoport, F. Furche, J. Chem. Phys. 133 (13) (2010) 134105. [45] X. Zhou, D. Sundholm, T.A. Wesoowski, V.R.I. Kaila, J. Am. Chem. Soc. 136 (7) (2014) 2723–2726. [46] J.E.D. Bene, J.D. Watts, R.J. Bartlett, J. Chem. Phys. 106 (14) (1997) 6051–6060. [47] M.J.G. Peach, P. Benfield, T. Helgaker, D.J. Tozer, J. Chem. Phys. 128 (4) (2008) 044118. [48] C. Bulliard, M. Allan, G. Wirtz, E. Haselbach, K.A. Zachariasse, N. Detzer, S. Grimme, J. Phys. Chem. A 103 (39) (1999) 7766–7772.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Natural transition orbitals for the calculation of correlation and excitation energies Sebastian Höfener ⇑, Wim Klopper Institute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), P.O. Box 6980, D-76049 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 16 February 2017 In final form 24 April 2017 Available online 26 April 2017
a b s t r a c t We propose a scheme to reduce the virtual space in the second-order approximate coupled-cluster singles-and-doubles (CC2) method, allowing for an accurate description of both ground-state correlation and excitation energies. A set of natural virtual orbitals is constructed using the ground-state MP2 density as well as the configuration-interaction-singles (CIS) excitation vectors. The results show that approximately half of the virtual space can be removed while an accuracy of about 90% for conventional correlation energies is obtained. CC2 excitation energies show in most cases an error of about 2% or less when the virtual space is reduced to half. Ó 2017 Published by Elsevier B.V.
1. Introduction Due to the scaling of correlation methods with basis-set size, many attempts have been made to truncate the size of the virtual orbital space not only for ground-state methods but also for response-property calculations. While first ideas concerning natural orbitals reach back to the 1950s [1–6], in 1997 it was shown by Klopper et al. that natural orbitals based on the second-order Møller-Plesset (MP2) density can provide >90% of the groundstate coupled-cluster singles and doubles (CCSD) correlation energy when using the occupation numbers of the natural virtual orbitals to remove half of the virtual space [7]. Some years later excitation vectors obtained from the configuration-interaction-sin gles (CIS) method were used to set up domains for coupled-cluster methods using localized orbitals [8,9]. Pair-natural orbitals (PNOs) [10–14] and orbital-specific virtual (OSV) orbitals have been (re) discovered and applied by many authors more recently to coupled-cluster methods [15–20] as well as to the random-phase approximation [21]. These methods are able to significantly reduce the virtual space and thus yield dramatically reduced computation times. However, a drawback of several methods is that the efficiency comes at the cost of non-orthonormal orbital sets for different orbital pairs and excited states, requiring special implementations. To circumvent this, a restricted-virtual space (RVS) approach was suggested by Send et al. who reduced the virtual space drastically using only the canonical virtual orbitals with orbital energies below 50 eV [22]. This method provides accurate ⇑ Corresponding author. E-mail addresses: [email protected] (S. Höfener), [email protected] (W. Klopper). http://dx.doi.org/10.1016/j.cplett.2017.04.083 0009-2614/Ó 2017 Published by Elsevier B.V.
excitation energies in many cases but ground-state correlation energies as well as frequency-dependent dipole polarizabilities using coupled-cluster theory can exhibit large errors [23] and it can lead to increased errors for np or pr states using ADC(2) [24]. In the present work, we construct effective natural transition orbitals (NTOs) suited for both ground and excited states via combining the ground-state MP2 density with CIS excitation vectors and a subsequent diagonalization of the virtual-virtual block of the so-obtained Fock matrix to form canonical orbitals. To the best of our knowledge, such a scheme is applied for the first time to calculate G0W0 quasi-particle energies. As an excited state property, we investigate oscillator strengths obtained at the second-order approximate coupled-cluster singles-and-doubles (CC2) level of theory [25]. 2. Method In order to reduce the space of virtuals efficiently, the key step is to combine appropriate measures for ground-state correlation energies and excitation energies, from which those virtual orbitals are extracted which belong to occupation numbers above a certain threshold. In the following, we discuss three methods to construct effective virtual spaces suited for both ground and excited states with a subsequent diagonalization of the so-obtained Fock matrix to form canonical orbitals. 2.1. Ground-state density For coupled-cluster methods using single and double excitations, it was shown by Klopper et al. that correlation calculations using only the natural virtual orbitals of the unrelaxed virtual-
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
virtual block of the MP2 ground-state density matrix (abbreviated ‘‘MP2 density matrix” throughout the present work),
DMP2 ¼ ab
X
DMP2;kern ; ij;abc
ð1Þ
ijc
with large occupation numbers yield a large fraction of the correlation energy [7]. The kernel is given as
DMP2;kern ¼ ij;abc
2ðiajg 12 jjcÞ ðicjg 12 jjaÞ 2ðibjg 12 jjcÞ : ac ij bc ij
ð2Þ
Here, g 12 denotes the Coulomb operator and pq is the sum of orbital energies of orbitals p and q, respectively. In order to reduce the scaling, it has also been proposed to use the diagonal contributions only [16],
DdMP2 ¼ ab
X MP2;kern Dii;abc ;
ð3Þ
53
and from the full set those vectors are collected that belong to singular values above a given threshold, while singular values belonging to a singular value below the threshold are removed. Note that an upper bound of the rank of the matrix RðnÞ is given by minðnvir ; noct Þ, that is, the minimum value of the number of virtual and (active) occupied orbitals. To give an example, for the helium atom only one particle NTO can be obtained, independent of the size of the basis. These excited-state NTOs are combined with a set of NVOs based on the MP2 density matrix,
X ¼ XMP2 Vð1Þ Vð2Þ . . . Vðnexc Þ :
ð9Þ
~ is calculated and linear For this set of orbitals the overlap S dependencies are removed using SVD,
~¼U ~R ~T; ~V XT X ¼ S
ð10Þ
~ which belong to singular values by deleting those vectors from V
ic
but, after observing a significant deterioration in accuracy, the full MP2 density matrix is used throughout the present work.
smaller than a fixed threshold of 108 to avoid numerical insta~ ~ together with an bilities, which yields the rectangular matrix V
2.2. Excited-state density
~ A linear-independent set of ~ appropriate eigenvalue matrix R. vectors is then obtained as
In the following, we discuss three main approaches, where for all schemes it is assumed that the CIS excitation vectors ð1Þ
~~ ~~ 1=2 Y ¼ XV R :
ð11Þ
ðnexc Þ
R ...R for the lowest nexc excitations in the full virtual space are available if needed. 1. (Augmented) RVS: Use the RVS approximation as proposed by Send et al. [22] and include additionally the natural virtual orbitals (NVOs) obtained from a diagonalization of the full MP2 density matrix,
XTMP2 DMP2 XMP2 ¼ d;
ð4Þ
where d is a diagonal matrix containing the eigenvalues and XMP2 are the eigenvectors. From the full set (XMP2 ) those eigenvectors (YMP2 ) are collected that belong to eigenvalues above a given threshold, while eigenvectors belonging to an eigenvalue below the threshold are removed. Finally, the two sets are joined and linear dependencies are removed. 2. Natural effective-density based orbitals (NEO): Using the ground-state MP2 density and an effective excited-state(s) density,
Dex ab ¼
nexc X X ðnÞ ðnÞ Rai Rbi ; n¼1
Finally, the orbitals obtained from the three schemes are rotated such that they diagonalize the virtual-virtual block of the Fock matrix, so that they can be used in a standard coupledcluster implementation assuming canonical orbitals provided that virtual orbitals can be frozen, i.e., omitted. 3. Results The proposed schemes have been implemented in the KOALA program package [26]. For all methods, the resolution-of-identity (RI) is used. First, the MP2-based NVOs are applied to the calculation of different ground-state correlation schemes to illustrate their broad applicability and to validate our implementation. Second, CC2 excitation energies and oscillator strengths are investigated using the new orbitals. All quantum-chemical methods used in the following are well-known in the literature, in the present work only the performance of the natural orbitals as proposed is assessed.
ð5Þ
i
3.1. Ground-state correlation energies
an effective density matrix is constructed using appropriate weights:
Deff ¼ wðMP2Þ DMP2 þ wðexÞ Dex : In the present work, the weights w
ð6Þ ðMP2Þ
ðexÞ
and w
were empiri-
cally chosen to be 1. The effective density matrix Deff is then diagonalized,
XT Deff X ¼ d;
ð7Þ
where d is a diagonal matrix containing the eigenvalues and X are the eigenvectors. From the full set (X) those eigenvectors (Y) are collected that belong to eigenvalues above a given threshold, while eigenvectors belonging to an eigenvalue below the threshold are removed. 3. Natural transition orbitals (NTOs): A singular value decomposition (SVD) is applied to each of the lowest nexc CIS excitation vectors, T
RðnÞ ¼ UðnÞ RðnÞ VðnÞ ;
ð8Þ
We begin with investigating the performance of NVOs for frozen-core RI-MP2 and RI-CCSD correlation energies calculated for the water molecule using the cc-pVXZ-F12 (X = D,T,Q) basis sets [27], see Table 1. The table shows that already small subsets can yield large fractions of the correlation energy. For example, for the cc-pVQZ-F12 basis a threshold of 1 104 yields 40 natural virtual orbitals, which corresponds to 27%, providing 94% (95%) of the MP2 (CCSD) correlation energy, which is similar to earlier work [7]. A striking efficiency of the MP2 NVOs is observed when the explicitly-correlated RI-MP2-F12/3B-sp method [28] is used. For MP2 the conventional convergence with respect to the basis is observed while for MP2-F12/3B-sp quasi-converged results are obtained for the double-zeta basis using the MP2 NVOs. For example, the result obtained in the cc-pVDZ-F12 basis employing NVOs with a threshold of 5 104 (17 virtual orbitals) corresponds to > 96% of the best value computed in the cc-pVQZ-F12 basis without truncation of the virtual space (150 virtual orbitals). This is rooted in the large redundancy of the conventional and the explicitly-correlated double excitations [29]. One comment on
54
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 1 Frozen-core correlation energies in mEh of the water molecule (d(O–H) = 1.8168 Bohr, a(HOH) = 104.109°) as calculated using the KOALA program using different basis sets. Values in parentheses are the number of virtual orbitals. Approx.e
Method RI-MP2
MP2 NVO
Thr.x 4
1 104
5 10
RI-MP-2F12
MP2 NVO
MP2 NVO
b c d e x
y
224.4 (17)
239.9 (20)
243.8 (21)
236.5 (25)
262.7 (33)
271.1 (40)
241.5 (43)
273.3 (84)
287.2 (150)
290.4 (17)
292.2 (20)
293.1 (21)
1 104
293.4 (25)
296.6 (33)
297.4 (40)
–
296.1 (43)
299.5 (84)
300.5 (150)
5 104
230.2 (17)
246.6 (20)
250.7 (21)
1 104
242.3 (25)
269.0 (33)
277.1 (40)
–
246.7 (43)
277.6 (84)
289.4 (150)
RI-CCSD a
cc-pVQZ-F12c
–
d
RI-CCSD
cc-pVTZ-F12b
5 104
RI-MP2 RI-MP2-F12d
cc-pVDZ-F12a
Geminal exponent c ¼ 0:9. Geminal exponent c ¼ 1:0. Geminal exponent c ¼ 1:1. RI-MP2-F12/2B-sp method, see Ref. [28]. Approximation for the virtual space. Threshold for removing virtual orbitals. Ref. [46].
the projector used in F12 theory is necessary at this point. When the virtual space is truncated, virtual orbitals might be removed from the orbital set that are not covered by the complementary auxiliary basis set (CABS). In order to avoid ‘‘holes” in the unified set of orbital basis and CABS, all basis functions belonging to the orbital basis are included in the CABS space. In case of a full virtual space the redundancy is eliminated automatically by the SVD orthogonalization during the CABS construction [30] while for truncated virtual spaces the removed virtual space is included in the CABS. A method which has experienced some attention in the past is given by the random-phase approximation (RPA), which can be obtained in a sum-over-states manner when all excited states are computed [31–34]. The results for correlation energies calculated using RPA are collected in Table 2, employing Eqs. (27) and (28) of Ref. [34],
X 1 X 0 xTDHF x0CIS ð12Þ 2 n o X X X X 1 ¼ x0TDHF x0CIS þ n x1TDHF x1CIS ; ð13Þ 4
EdRPA ¼ c ErCCD c
P 0 P where x and x1 denote the sums of the singlet and triplet excitation energies of the method given in the subscript, respecflip flip while n ¼ 3 yields ErCCD;w [34]. Note tively; n ¼ 1 yields ErCCD;w=o c c that the spin-flipped excitations do not contribute to the dRPA cor-
relation energy. Although the absolute values of the sums of the excitation energies are changed by hundreds of hartree when NVOs are used, the final correlation energy shows an accuracy of about 85% when only 38% of the virtual orbitals are included. When (vertical) ionization energies are to be studied, the manybody Green’s function GW formalism has been shown to provide reliable results, which has been applied to molecules recently [35,36]. In this method, quasi-particle (QP) energies are calculated as Kohn-Sham (KS) orbital energies shifted using, e.g., TDA or RPA excitation vectors according to KS KS KS QP p ¼ p þ Z p h/p jRX þ RC ðp Þ V xc j/p i:
ð14Þ
Note that using NVOs, the truncated virtual space is used for the excited states only, while all other terms are computed using the original, that is, untruncated Kohn-Sham MOs, see appendix. Results for this method are collected in Table 3. To give an example, for the rpa-G0W0 method 14 NVOs yield a quasi-particle energy QP ðr1s Þ of 403:5 eV, while the final result using 55 virtual orbitals is 402:2 eV, exhibiting an error of 1:2 eV. Tightening the threshold to 1 104 decreases the error to below 0:3 eV. For the highest occupied molecular orbital (HOMO), QP ðr2p Þ, a reduced dependence on the virtual space is observed using NVOs, for which the biggest error is below 0:05 eV for rpa-G0W0, while for the RVS method the error amounts about 0:3 eV. A different behavior is
Table 2 All-electron RI-RPA and RI-dRPA correlation energies in Eh of the N2 molecule (d(N–N) = 2.0749 bohr), cf. Ref. [34]. All calculations employ a def2-TZVPP basis with corresponding auxiliary basis sets. All energies are based on a Hartree-Fock determinant and MP2 natural orbitals as obtained from the full virtual-virtual block of the MP2 density. Method Thr.b nvir a P 0
xTDHF x1 P TDHF x0 P CIS x1CIS P
Without flipc With flipd a b c d
RI-dRPA 4
5 10 21 (147) 1026.20
4
1 10 46 (322) 3017.49
RI-RPA
55 (385) 3555.64
5 10 21 (147) 942.53
1 104 46 (322) 2833.28
55 (385) 3346.33 3327.41
Full
4
Full
1016.95
3000.06
3537.50
932.71
2815.08
1026.90
3018.41
3556.57
943.18
2834.02
3347.08
1016.95
3000.06
3537.50
933.23
2815.68
3328.02
0.35
0.46
0.47
0.29 0.55
0.34 0.64
0.34 0.65
Number of virtual orbitals; parentheses: number of excitation vectors. Threshold for removing virtual orbitals. Without spin-flipped (de) excitations. With spin-flipped (de) excitations.
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S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 3 All-electron PBE G0W0 quasi-particle energies QP in eV of the N2 molecule (d(N–N) = 2.0749 bohr), cf. Ref. [34]. All calculations employ a def2-TZVPP basis with corresponding auxiliary basis sets and grid 5 for the DFT integration. MP2 natural orbitals as obtained from the full virtual-virtual block of the MP2 density. The damping parameter d was set to zero in all calculations. Method Thr.b nvir a
QP ðr2p Þ QP ðp2p Þ QP ðr2p Þ QP ðp2p Þ QP ðr2s Þ QP ðr2s Þ QP ðr1s Þ QP ðr1s Þ a b c
tda-G0W0
rpa-G0W0
1 103 14 (98)
5 104 17 (119)
1 104 46 (322)
RVSc
Full
5 104 17 (119)
1 104 46 (322)
RVSc
Full
55 (385)
1 103 14 (98)
22 (154)
22 (154)
55 (385)
8.7
8.7
8.6
3.4
3.2
2.8
8.5
8.4
9.1
9.0
8.9
8.9
8.8
3.1
2.6
3.5
3.4
3.0
3.2
14.0
14.1
2.8
14.0
13.7
14.0
14.8
14.9
14.8
14.5
14.8
16.8 16.8
16.9
16.7
16.3
16.6
16.4
16.4
16.3
15.9
16.3
16.8
16.7
16.3
16.6
17.9
17.9
17.8
17.5
17.8
29.4 397.4
29.4 397.2
30.0 396.1
29.8 398.2
29.9 395.8
30.7 403.4
30.7 403.3
30.8 402.4
29.0 403.9
29.0 402.1
397.4
397.3
396.2
398.2
395.9
403.5
403.4
402.5
403.9
402.2
Number of virtual orbitals; parentheses: number of excitation vectors. Threshold for removing virtual orbitals. Threshold for orbital energies: 50 eV.
observed for the lowest unoccupied orbital (LUMO),
QP p2p
,
however. Here, NVOs lead to an error slightly above 0:2 eV employing a threshold of 1 104 , while RVS increases the error to about 0:5 eV and 0:4 eV for tda-G0W0 and rpa-G0W0, respectively. The table also reveals that no systematic improvement is obtained when additional natural orbitals are used, for example, in case of tda-G0W0 a threshold of 1 103 yields an error of 0:02 eV for the HOMO, while a threshold of 5 104 results in an error of about 0:1 eV. However, the errors introduced using truncated virtual spaces using a threshold of 5 104 remain for the occupied orbitals significantly below the method error in case of tda-G0W0 when the rpa-G0W0 values are taken as reference. For example, the truncated virtual space introduces an error of about 0:1 eV for the HOMO for tda-G0W0, while the difference of full-space tda-G0W0 and rpa-G0W0 is as large as 0:8 eV. In our test system, the only exception is the quasi-particle energy of the LUMO which exhibits an error of about 0:6 eV for a threshold of 5 104 compared to full-space tda-G0W0, while the difference of full-space tda-G0W0 and rpa-G0W0 amounts ca. 0:2 eV. 3.2. Natural orbitals for CC2 ground-state correlation and excitation energies Having investigated the performance of NVOs for which only the MP2 density was used, we now turn our attention to excited states, for which also further virtual orbitals are included to provide a balanced description for both ground and excited states. First, results for the benzene and DMABN molecule are collected in Tables 4 and 5, respectively, both calculated using frozen-core RICC2 [25,37] and Dunning’s aug-cc-pVTZ basis [38,39]. Both tables also reveal that both the original RVS and augmented RVS methods (applying a threshold of 50 eV) yield correlation energies which deviate significantly while excitation energies show a sufficient accuracy. Note that the large deviation for correlation energies is expected for RVS. Concerning higher excited states, both tables also reveal that the NEO approach seems to yield the largest errors of about 0.2 eV, but the virtual space is by far the smallest. Furthermore, the (augmented) RVS method seems to introduce small errors for oscillator strengths for DMABN for the CT excitation. Second, the ethylene molecule is investigated with respect to the 11 B3u and 21 B3u excited states exhibiting valence and Rydberg character, respectively. The Rydberg basis ‘‘CM2”, which was added to the standard def2-TZVPPD basis, consists of 2s2p2d with exponents 0:01 and 0:0033 [40], respectively, which are placed in the
center of mass of the molecule [41]. It was shown that the CIS method, which is used in the present work as a prescreening method, is able to capture the Rydberg state of the ethylene molecule if a proper Rydberg basis is included [42]. While the conventional basis yields virtual orbitals with orbital energies of about 0.1 eV and higher, the CM2 basis provides virtual orbitals with orbital energies in the between the eigenvalue of the HOMO and about 0.1 eV. Results for RI-CC2 ground-state correlation energies and excitation energies are collected in Table 6. The table confirms that the original RVS approach seems to yield rather large errors in the oscillator strengths and that the NEO approach yields by far the smallest virtual space. It also illustrates how different the requirements for ground-state correlation energies and (Rydberg) excitations can be: While the ground-state correlation energy shows an error of about 0.1% using a threshold of 1 105 , the Rydberg excitation exhibits an error of more than 4 eV (44%) when only the ground-state MP2 density is used. This large error arises because the diffuse functions are not important for the ground-state correlation energy and are thus not included even for a rather tight threshold of 1 105 , highlighting the need to include measures for individual excited states. Note, however, that the Rydberg excitation is the twelfth excitation and thus at least 12 excitations have to be calculated using C 1 symmetry. Using symmetry, only the lowest two excitations in B3u would be calculated, reducing the virtual space even further. Such an additional reduction was achieved by taking into account the MP2 density and the excitation vector of the Rydberg state only, see Table 6, leading to virtual spaces that are well suited for ground-state correlation energies and the target Rydberg state showing errors of about 5% and 0.1 eV, respectively, while the valence state, which was not taken into account for the determination of the reduced virtual space, exhibits errors of 0.4 eV and larger. Finally, RI-CC2 calculations using the Karlsruhe def2-SVPD and def2-TZVP basis sets [43,44] are carried out for the lowest singlet state (S1 ) of deprotonated retinal in vacuum, of which the geometry displayed in Fig. 1 was taken from Ref. [45]. Data including the original RVS method of Send et al. is collected in Table 7. It confirms that the (augmented) RVS-based method performs well for the excitation energies in almost all cases but can yield large deviations in the correlation energies even for rather tight thresholds, cf. Ref. [23]. To give an example, while RVS using a 103 threshold yields 534 virtual orbitals and a correlation energy of 3.112 Eh corresponding to 84%, NEO using a threshold of 104 yields 523 virtual orbitals and a correlation energy of 3.673 Eh corresponding to 99%. This again illustrates how important the
56
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 4 The first (pp B2u ), fifth (pr A2u ), and tenth (pp E1g ) excitation energies in eV of the benzene molecule (d(C–C) = 2.6399 Bohr, d(C–H) = 2.0485 Bohr) as calculated using the KOALA program employing Dunning’s aug-cc-pVTZ basis. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Method RI-CC2
c d e x y
Thr.x
nvir c
Corr.a
pp B2u b
pr A2u b
pp E1g b
1 5 104
187 246
639.1 855.9
5.32 (0.00) 5.25 (0.00)
6.71 (0.07) 6.94 (0.07)
7.12 (0.63) 7.13 (0.66)
1 104
272
902.7
5.24 (0.00)
6.98 (0.07)
7.13 (0.66)
99
858.9
5.29 (0.00)
6.91 (0.07)
7.33 (0.71)
1 104
174
944.2
5.24 (0.00)
6.99 (0.07)
7.21 (0.70)
5 104
198
892.4
5.24 (0.00)
6.92 (0.07)
7.13 (0.66)
1 104
297
962.3
5.22 (0.00)
6.98 (0.07)
7.13 (0.66)
RI-CC2
–
393
972.9
CCSDy
–
RI-CC2
a
(a) RVS
e
5 104
RI-CC2
b
Approx.d
NEO
NTO
5.22 (0.00)
6.99 (0.06)
7.12 (0.66)
5.14 (0.00)
6.99 (0.07)
7.29 (1.33)
Correlation energy in mEh . Excitation energy in eV. Number of virtual orbitals. Approximation for the virtual space. Threshold for orbital energies: 50 eV. Threshold for removing virtual orbitals. Ref. [46].
Table 5 The first (1 B) and second (1 A) excitation energy in eV of the 4-(N; N-dimethylamino) benzonitrile (DMABN) molecule as calculated using the KOALA program employing Dunning’s aug-cc-pVTZ basis. The geometry was taken from Ref. [47]. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Thr.x
nvir c
Corr.a
1 5 104
321 426
1.219 1.672
4.42 (0.03) 4.41 (0.03)
4.46 (0.15) 4.59 (0.32)
1 104
478
1.762
4.39 (0.03)
4.61 (0.37)
4
5 10
166
1.676
4.48 (0.02)
4.71 (0.40)
1 104
302
1.843
4.42 (0.02)
4.67 (0.46)
5 104
199
1.638
4.44 (0.03)
4.62 (0.30)
1 104
374
1.857
4.41 (0.03)
4.65 (0.44)
RI-CC2
–
697
1.898
4.40 (0.03)
4.64 (0.45)
Exp.y
–
4.25
4.56
Method RI-CC2
RI-CC2
RI-CC2
a b c d e f x y
Approx.d (a) RVS
NEO
NTO
f
1
B (L)b,e
1
A (CT)b,e
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Approximation for the virtual space. Local (L) and charge-transfer (CT) excitation, cf. Ref. [47]. Threshold for orbital energies: 50 eV. Threshold for removing virtual orbitals. Ref. [48].
MP2 density is for a balanced description of ground and excited states. Results for the NEO and NTO approach are collected in Tables 8 and 9, respectively. Both approaches are related to each other and thus yield similar but not identical results. In case of NEOs, 359 virtual orbitals for the def2-SVPD basis (threshold 5 104 ) yield a correlation energy of 3.510 Eh , while 368 NTOs (threshold 1 103 ) yield a correlation energy of 3.391 Eh . On the other hand, for this example NEOs yield an excitation energy of 3.72 eV while NTOs yield 3.68 eV. Both formulations reduce the virtual space to about half of its original size for the cases investigated, while NTOs seem to perform slightly better than NEOs for excitation energies while NEOs seem to perform slightly better than NTOs for correlation energies.
4. Conclusions The investigations presented show that simplistic schemes to truncate the virtual space can lead to results that are biased either
to the ground state or to excited states. A pragmatic solution is to combine information from different estimates and span a subspace to find a compromise, corresponding to virtual orbitals obtained from the MP2 ground-state density and virtual orbitals obtained from CIS excitation vectors. This approach can be formulated in two ways, either via an effective density or via natural transition orbitals with subsequent orthonormalization. Both formulations reduce the virtual space to about half of its original size for the cases investigated, while NTOs seem to perform slightly better than NEOs for excitation energies. On the other hand, NEOs seem to perform slightly better than NTOs for correlation energies. However, the CIS treatment must provide a good zeroth-order description for both methods which is the case for HOMO-LUMO transitions, for instance. Concerning the calculation of quasi-particle energies, the use of NVOs shows a varying performance depending on the molecular orbital under investigation. While for the N2 molecule the errors are 0.1 eV or less for the HOMO, the errors for other orbitals can increase to more than 1 eV using truncated virtual spaces. Although a single test case is not able to validate whether the accu-
57
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 6 11 B3u (valence state) and 21 B3u (Rydberg state) excitation energy in eV of the ethylene (C2H4) molecule employing the def2-TZVPPD + CM2 basis, see main text. The geometry was taken from Ref. [42]. Values in parentheses are the oscillator strengths. A threshold of 1 corresponds to the original RVS method [22]. Method/ Approx.d RI-CC2/MP2
RI-CC2/(a) RVS
Thr.a
nvir b
Corr.c
11 B3u (valence)
21 B3u (Rydberg)
1 103
24
277.6
10.31 (0.437)
19.34 (0.153)
1 104
65
364.5
8.90 (0.424)
15.22 (0.128)
1 105
109
375.1
8.51 (0.414)
13.24 (0.088)
1
80 95
215.9 285.9
7.95 (0.364) 8.01 (0.294)
8.91 (0.046) 9.04 (0.036)
1 103
RI-CC2/NEO
RI-CC2/NTO
RI-CC2/NEO
RI-CC2/NTO
1 104
107
318.7
8.03 (0.377)
9.11 (0.031)
1 105
122
341.4
8.03 (0.379)
9.15 (0.028)
1 103
38
279.0
8.21 (0.381)
9.07 (0.067)
1 104
79
364.0
8.09 (0.387)
9.16 (0.032)
1 105
126
375.2
8.04 (0.382)
9.17 (0.026)
3
101
322.7
8.03 (0.378)
9.11 (0.030)
1 104
135
372.5
8.03 (0.380)
9.17 (0.027)
1 105
152
375.6
8.03 (0.381)
9.17 (0.026)
1 104
66
363.9
8.81 (0.385)
9.18 (0.052)
1 105
114
375.2
8.42 (0.393)
9.18 (0.031)
1 104
73
365.3
8.79 (0.388)
9.17 (0.048)
1 105
117
375.2
8.42 (0.392)
9.18 (0.031)
–
152
375.6
8.03 (0.381)
9.17 (0.026)
1 10
x
x
RI-CC2 a b c d x
Threshold for removing virtual orbitals. Number of virtual orbitals. Correlation energy in mEh . Approximation for the virtual space. Only the MP2 density and the excitation vector for the Rydberg excitation are used.
Fig. 1. Ground-state geometry of the deprotonated rhodopsin as taken from Ref. [45]. Nitrogen atoms are plotted in blue, carbon atoms in gray, and hydrogen atoms in white. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 7 The first excitation energy in eV of the rhodopsin system at RI-CC2/(a) RVS level of theory as calculated using the KOALA program. The threshold for orbital energies is 50 eV. A threshold of 1 corresponds to the original RVS method [22]. Thr.x
def2-SVPD nvir
1
a b c x
c
a
def2-TZVP Corr.a
S1b
340 534
2.691 3.609
3.64 3.62
Corr.
S1b
nvir
c
1 103
456 534
2.647 3.112
3.70 3.67
5 104
574
3.252
3.66
620
3.842
3.61
1 104
647
3.516
3.65
671
4.111
3.61
1 105
695
3.670
3.65
848
4.142
3.60
–
702
3.694
3.65
903
4.178
3.60
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
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S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
Table 8 The first excitation energy in eV of the rhodopsin system at RI-CC2/NEO level of theory as calculated using the KOALA program. Thr.x
def2-SVPD nvir
a b c x
c
def2-TZVP S1b
a
Corr.
Corr.a
S1b
297
3.499
3.72
391
3.757
3.68
671
4.111
3.62
3.65
878
4.176
3.60
3.65
903
4.178
3.60
3
1 10
277
3.296
3.76
5 104
359
3.510
3.72
1 104
523
3.673
3.66
1 105
682
3.694
–
702
3.694
nvir
c
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
Table 9 The first excitation energy in eV of the rhodopsin system at RI-CC2/NTO level of theory as calculated using the KOALA program. Thr.x
def2-SVPD nvir
a b c x
c
def2-TZVP
a
S1b
Corr.
nvir
b
Corr.a
S1b
3
1 10
368
3.391
3.68
388
3.610
3.64
5 104
460
3.576
3.66
495
3.848
3.62
1 104
642
3.688
3.65
793
4.147
3.61
1 105
702
3.694
3.65
903
4.178
3.60
–
702
3.694
3.65
903
4.178
3.60
Correlation energy in Eh . Excitation energy in eV. Number of virtual orbitals. Threshold for removing virtual orbitals.
racy obtained using NVOs is sufficient, the numerical example reveals that a simple truncation of the virtual space using RVS can lead to large error bars in particular for the HOMO, from which the ionization energy is to be deduced, and does thus not suffice. We expect a similar performance of NVOs and RVS for advanced GW schemes beyond the G0W0 approximation.
h/KS p j RC
XX j/KS KS p i ¼ p
2 KS j /KS p /k jqn j
n
k
þ
KS KS p k þ xn id
2 KS j /KS p /c jqn j
XX c
n
KS KS p c xn þ id
;
ð18Þ
where n runs over all excited states, and thus
Acknowledgments S.H. gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft DFG (HO-4605/2-1)
D E XX KS /KS j/ ¼ p jð@ RC ðÞ=@ Þ¼KS p p k
Appendix A
n
2 KS j /KS p /k jqn j
KS KS p k þ xn id
XX
2
2 KS jð/KS p /c jqn Þj KS ðKS p c xn þ idÞ
2
c
n
:
A.1. Equations used for G0W0
ð19Þ
In the G0W0 method using natural orbitals, the quasi-particle energies are calculated according to:
The natural orbitals only enter the calculation of the excitedstates transition densities:
KS KS KS QP p ¼ p þ Z p h/p jRX þ RC p
jqn Þ ¼
V xc j/KS p i;
ð15Þ
KS where /KS p and p are orbitals and orbital energies of the original basis, respectively. This leads to the following working equations:
D
E X KS KS KS KS KS /KS /p /i j/i /p ¼ p jRX j/p
ð16Þ
i
n o1 j/KS : Z p ¼ 1 h/KS p jð@ RC ðÞ=@ Þ¼KS p i p The correlation part of the self energy is computed as:
ð17Þ
X NO NO NO 1=2 ðnÞ j/b /j ÞðNO Z bj x1=2 ; b j Þ n
ð20Þ
bj
obtained from the RPA eigenvalue problem in the truncated (NO) space:
M ZðnÞ ¼ ZðnÞ x2n ;
ð21Þ
where
M ¼ ðA BÞ1=2 ðA þ BÞ ðA BÞ1=2 :
ð22Þ
The (closed-shell) orbital-rotation Hessians are given as:
Aai;bj ¼
NO NO a i
Bai;bj ¼ 2ðaNO i
NO
NO
NO NO
dij dab þ 2ðaNO i jb j Þ
NO NO
jb j Þ:
ð23Þ ð24Þ
S. Höfener, W. Klopper / Chemical Physics Letters 679 (2017) 52–59
References [1] P.-O. Löwdin, Correlation problem in many-electron quantum mechanics I. review of different approaches and discussion of some current ideas, in: I. Prigogine (Ed.), Advances in Chemical Physics, vol. 2, John Wiley & Sons, Inc., Hoboken, NJ, USA, 1958, http://dx.doi.org/10.1002/9780470143483.ch7. [2] W. Kutzelnigg, Theoret. Chim. Acta 1 (4) (1963) 327–342. [3] C.F. Bender, E.R. Davidson, J. Phys. Chem. 70 (8) (1966) 2675–2685. [4] C.F. Bender, E.R. Davidson, J. Chem. Phys. 47 (12) (1967) 4972–4978. [5] C.F. Bender, E.R. Davidson, Phys. Rev. 183 (1969) 23–30. [6] S.A. Houlden, I.G. Csizmadia, Theoret. Chim. Acta 30 (3) (1973) 209–216. [7] W. Klopper, J. Noga, H. Koch, T. Helgaker, Theor. Chem. Acc. 97 (1997) 164. [8] T. Korona, H.-J. Werner, J. Chem. Phys. 118 (7) (2003) 3006–3019. [9] D. Kats, T. Korona, M. Schütz, J. Chem. Phys. 125 (10) (2006) 104106. [10] W. Meyer, Int. J. Quant. Chem. 5 (S5) (1971) 341–348. [11] W. Meyer, J. Chem. Phys. 58 (3) (1973) 1017–1035. [12] R. Ahlrichs, H. Lischka, V. Staemmler, W. Kutzelnigg, J. Chem. Phys. 62 (4) (1975) 1225–1234. [13] R. Ahlrichs, F. Driessler, H. Lischka, V. Staemmler, W. Kutzelnigg, J. Chem. Phys. 62 (4) (1975) 1235–1247. [14] R. Ahlrichs, F. Keil, H. Lischka, W. Kutzelnigg, V. Staemmler, J. Chem. Phys. 63 (1) (1975) 455–463. [15] F. Neese, F. Wennmohs, A. Hansen, J. Chem. Phys. 130 (11) (2009) 114108. [16] J. Yang, Y. Kurashige, F.R. Manby, G.K.L. Chan, J. Chem. Phys. 134 (4) (2011) 044123. [17] C. Krause, H.-J. Werner, Phys. Chem. Chem. Phys. 14 (2012) 7591–7604. [18] D.P. Tew, C. Hättig, Int. J. Quant. Chem. 113 (3) (2013) 224–229. [19] B. Helmich, C. Hättig, J. Chem. Phys. 135 (21) (2011) 214106. [20] G. Schmitz, B. Helmich, C. Hättig, Mol. Phys. 111 (16–17) (2013) 2463–2476. [21] M. Kállay, J. Chem. Phys. 142 (20) (2015) 204105. [22] R. Send, V.R.I. Kaila, D. Sundholm, J. Chem. Phys. 134 (21) (2011) 214114. [23] A. Kumar, T.D. Crawford, J. Phys. Chem. A 121 (3) (2017) 708–716. [24] C. Yang, A. Dreuw, J. Comput. Chem. (2017), http://dx.doi.org/10.1002/ jcc.24794.
59
[25] O. Christiansen, H. Koch, P. Jørgensen, Chem. Phys. Lett. 243 (5–6) (1995) 409– 418. [26] S. Höfener, J. Comput. Chem. 35 (23) (2014) 1716–1724. [27] K.A. Peterson, T.B. Adler, H.-J. Werner, J. Chem. Phys. 128 (8) (2008) 084102. [28] R.A. Bachorz, F.A. Bischoff, A. Glöß, C. Hättig, S. Höfener, W. Klopper, D.P. Tew, J. Comput. Chem. 32 (11) (2011) 2492–2513. [29] S. Höfener, D.P. Tew, W. Klopper, T. Helgaker, Chem. Phys. 356 (13) (2009) 25–30. [30] E.F. Valeev, Chem. Phys. Lett. 395 (46) (2004) 190–195. [31] Z. Yan, J.P. Perdew, S. Kurth, Phys. Rev. B 61 (2000) 16430–16439. [32] F. Furche, Phys. Rev. B 64 (2001) 195120. [33] F. Furche, J. Chem. Phys. 129 (11) (2008) 114105. [34] W. Klopper, A.M. Teale, S. Coriani, T.B. Pedersen, T. Helgaker, Chem. Phys. Lett. 510 (13) (2011) 147–153. [35] M.J. van Setten, F. Weigend, F. Evers, J. Chem. Theory Comput. 9 (1) (2013) 232–246. [36] K. Krause, M.E. Harding, W. Klopper, Mol. Phys. 113 (13-14) (2015) 1952– 1960. [37] C. Hättig, F. Weigend, J. Chem. Phys. 113 (13) (2000) 5154–5161. [38] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [39] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [40] O. Christiansen, H. Koch, A. Halkier, P. Jorgensen, T. Helgaker, A. Sánchez de Merás, J. Chem. Phys. 105 (16) (1996) 6921. [41] K. Kaufmann, W. Baumeister, M. Jungen, J. Phys. B: At. Mol. Opt. Phys. 22 (14) (1999) 2223–2240. [42] J.S. Kottmann, S. Höfener, F.A. Bischoff, Phys. Chem. Chem. Phys. 17 (2015) 31453–31462. [43] F. Weigend, M. Häser, H. Patzelt, R. Ahlrichs, Chem. Phys. Lett. 294 (1998) 143. [44] D. Rappoport, F. Furche, J. Chem. Phys. 133 (13) (2010) 134105. [45] X. Zhou, D. Sundholm, T.A. Wesoowski, V.R.I. Kaila, J. Am. Chem. Soc. 136 (7) (2014) 2723–2726. [46] J.E.D. Bene, J.D. Watts, R.J. Bartlett, J. Chem. Phys. 106 (14) (1997) 6051–6060. [47] M.J.G. Peach, P. Benfield, T. Helgaker, D.J. Tozer, J. Chem. Phys. 128 (4) (2008) 044118. [48] C. Bulliard, M. Allan, G. Wirtz, E. Haselbach, K.A. Zachariasse, N. Detzer, S. Grimme, J. Phys. Chem. A 103 (39) (1999) 7766–7772.