Electronic Excited States from the Adiabatic-Connection Formalism

Letter pubs.acs.org/JPCL

Cite This: J. Phys. Chem. Lett. 2018, 9, 5534−5538

Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wave Functions Ewa Pastorczak and Katarzyna Pernal* Institute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924 Lodz, Poland

J. Phys. Chem. Lett. 2018.9:5534-5538. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/30/18. For personal use only.

S Supporting Information *

ABSTRACT: It is demonstrated how the recently proposed multireference adiabaticconnection (AC) approximation for electron correlation energy (Pernal, K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions. Phys. Rev. Lett. 2018, 120, 013001) can be extended to predicting correlation energy in excited states of molecules. It is the first successful application of the AC approach to computing excitedstates energies of molecules using a complete active space (CAS) wave function as a reference. The unique feature of the AC-CAS approach with respect to popular methods such as CASPT2 and NEVPT2 is that it requires only one- and two-particle reduced density matrices, making it possible to efficiently treat large spaces of active electrons. Application of the simpler variant of AC, the AC0, which is based on the first-order expansion of the AC integrand at the uncorrelated system limit, to excited states yields excitation energies with accuracy rivaling that of the NEVPT2 method but at greatly reduced computational cost. Assume that a given reference state Ψref is an approximate representation of the Pth exact state. The error in the energy with respect to the exact value EP arising from using an approximate wave function will be called a correlation energy for the Pth state

U

nlike the ground states, for whose qualitative description a single-determinant wave function is usually sufficient, excited electronic states frequently manifest multireference character.1,2 As a consequence, to account for the static correlation, one needs to employ expensive methods based on multiconfiguration wave functions. At the same time, capturing the dynamic correlation is essential for obtaining quantitatively correct potential energy surfaces and electronic spectra.3,4 This task can be realized in several ways, the most popular being perturbation corrections, e.g., CASPT2 and NEVPT2.5−8 While the former method exhibits such problems as lack of size-consistency9 and intruder states,10 the main disadvantage of both approaches is the unfavorable scaling of the computational cost with the size of the system and with sophistication of the wave function. The need to calculate the three- and four-particle reduced density matrices11 (RDMs) causes the cost of computation to grow as the eighth power of the number of active orbitals in complete active space (CAS). Circumventing this requirement either through the use of cumulant approximations12,13 or linear-response theory14,15 has proven challenging. Alternatives to perturbation corrections include density functional theory approaches16,17 and quantum Monte Carlo methods.18 The former, while very efficient, possess a certain degree of empiricism, and the cost of the latter still remains very high. We present two efficient, nonempirical approaches to calculating excited-state energies using a CAS reference. It is shown that, while considerably more efficient in treating active orbitals than perturbation approaches, the proposed adiabaticconnection (AC)-CAS and AC0-CAS methods produce results on par with such sophisticated methods as NEVPT2 and CASPT2. © 2018 American Chemical Society

Ecorr = EP − ⟨Ψ ref |Ĥ |Ψ ref ⟩

(1)

where Ĥ is a nonrelativistic electronic Hamiltonian of a given system. In refs 19 and 20 we have proposed the AC correlation energy formula for ground states. It has been based on groupproduct partitioning of the Hamiltonian Ĥ = Ĥ (0) + Ĥ ′, where Ĥ (0) includes a two-particle interaction operator in the active space,21 and on defining the AC Hamiltonian as Ĥ α = Ĥ (0) + αĤ ′, with the coupling parameter α ranging from 0 to 1. The AC formula for the correlation energy has been expressed in terms of transition RDMs γα,ν, occupation numbers np, and two-electron integrals ⟨rs|pq⟩ in the following form AC Ecorr =

∫0

1

W α dα ij

where the AC integrand reads Wα =

1 2

(2)

yz

∑ ′jjjjj∑ γprα ,νγsqα ,ν + (np − 1)nqδrqδpszzzzz⟨rs|pq⟩ pqrs

k

ν

{

(3)

and a prime indicates exclusion of terms involving all four indices pqrs belonging to a set of either active or inactive orbitals. The AC expression has been obtained under an Received: August 4, 2018 Accepted: September 7, 2018 Published: September 7, 2018 5534

DOI: 10.1021/acs.jpclett.8b02391 J. Phys. Chem. Lett. 2018, 9, 5534−5538

Letter

The Journal of Physical Chemistry Letters assumption that a one-electron RDM stays constant along the AC path. Its extension to excited sates is straightforward, namely, if the reference wave function represents a Pth exact state, then transition RDMs connect the Pth and νth states of α † α ̂α α the AC Hamiltonian, i.e., γα,ν pq = ⟨ΨP|âq âp|Ψν ⟩ , where H Ψν = α α Eν Ψν . In the approximate AC formalism, transition RDMs are obtained as22 l o (n − nq)[Y αν ]pq ∀p > q o o p (γ α , ν)qp = m o o o(np − nq)[X αν ]qp ∀q > p n

[Xαν ,

When exploring the potential energy surface of a given state, a proper reference wave function should lead to a constant number of negatively normalized vectors in ERPA; otherwise, discontinuities may arise on the energy surface. One should be aware that for a stable reference wave function the number of negative excitations should be equal to P − 1.28 However, this number may be smaller than P − 1 due to the approximate nature of the ERPA polarization propagator. It is important to notice that for an excited-staterepresenting wave function the Hessian matrix is not positive-definite and the ( − ) and ( + ) matrices in general may have negative eigenvalues.28 Consequently, the ERPA problem in eq 5 can still be turned into a symmetric half-sized eigenproblem, but it would involve square roots of the above-mentioned matrices with imaginary eigenvalues. To avoid imaginary-valued matrices, the ERPA problem for excited states should be rather turned into a half-sized realvalue nonsymmetric eigenproblem reading

(4)

Yαν ]

where the vectors follow from the Extended Random Phase Approximation (ERPA) equations. ERPA provides extension of RPA for multireference wave functions,23,24 and it takes form of a generalized eigenproblem reading ij ( ) yzjij X ν zyz ij− 5 0 yzjij X ν zyz jj z j zzjj zz j ) ( zzjjj Y zzz = ωνjj 0 zj Y z 5 ν k {k { k {k ν {

(5)

[5 −1(( + ))5 −1(( − ))](Yν − X ν) = ων 2(Yν − X ν)

The matrices ( and ) are determined by the Hamiltonian and the reference wave function for a given system as follows ∀ p > q ( pq , rs = ⟨Ψ ref |[ap†̂ aq̂ , [Ĥ , aŝ†ar̂ ]]|Ψ ref ⟩ r>s

∀ p > q )pq , rs = ⟨Ψ r>s

ref

|[ap†̂ aq̂ ,

[Ĥ , ar̂ †aŝ ]]|Ψ ref ⟩

(11)

and (6)

(Yν + X ν) = (7)

(8)

r>s

Throughout the paper, indices pqrs correspond to natural spin orbitals of the reference function. It is worth noticing that in the ERPA approach excitation operators replacing one active (partially filled) orbital with another active orbital â†pâq, where 0 < np and nq < 1, are nonredundant. As a result, when orbitals are partitioned into subsets of inactive, np = 1, active, 0 < np < 1, and secondary (virtual), np = 0, orbitals, the allowed excitations in the ERPA approach involve inactive−secondary, inactive−active, active−active, and active−secondary pq pairs in â†pâq operators. Solutions of eq 5 come in pairs of positive and negative eigenvalues, ων+ and ων−, equal up to the absolute value, ων+ = −ων−, whereas the corresponding eigenvectors possess the property Yν− = Xν+ and Xν− = Yν+. This “pairing” of solutions of the ERPA equation results from the structure of the ERPA eigenproblem and occurs irrespective of whether Ψref represents a ground or an excited state.24,25 For a ground-state Ψref, all eigenvectors corresponding to positive eigenvalues ων+ would assume positive normalization20 T

[Yν+] 5Yν +

1 − [X ν+]T 5X ν+ = 2

dW α α dα α= 0

1 2

5 −1(( − ))(Yν − X ν) (12)

= W (0) + W (1)α , and noticing that W(0) = 0, which,

after exploiting the formula in eq 2, has resulted in the AC0 approximation (see the Supporting Information for details)

(9)

In the case of Ψref representing an excited state, one expects that in addition to eigenvectors normalized as shown in eq 9 there should exist vectors conforming to negative normalization [Yν+]T 5Yν+ − [X ν+]T 5X ν+ = −

ων

2

For a stable reference Ψref, the eigenvalues {ων2} are real and positive. ERPA eigenvectors [Xαν ,Yαν ] obtained from eqs 11 and 12 correspond to positive values of ων, and they should be normalized as shown in eqs 9 and 10. The presence of the negatively normalized eigenvectors (negative excitations) is related to de-excitation modes in the ERPA propagator; therefore, it is only expected if the reference wave function describes an excited state. Employing the ERPA approximation for the RDMs in the AC formalism allows one to obtain the correlation energy solely by means of one- and two-electron RDMs of the given wave function Ψref. Direct use of the AC formula involves solving the ERPA equations written for the AC Hamiltonian for a number of values of α. As has been discussed, ERPA corresponding to a full Hamiltonian Ĥ α=1 is expected to yield negative-excitation solutions, defined by eq 10, if Ψ ref represents an excited state. The same is true in the noncorrelated limit corresponding to Ĥ α=0 because, by construction, it is assumed that Ψref is the Pth state of the Ĥ (0) = Ĥ α=0 Hamiltonian. On the basis of the observation that the AC integral is typically nearly linear,19,20 the correlation energy has been further approximated using the first‑order expansion of Wα at α = 0, i.e., Wα = Wα=0 +

and the metric matrix is given by natural occupation numbers np (pertaining to Ψref) ∀ p > q 5pq , rs = δprδqs(nq − np)

1

AC0 Ecorr =

W (1) 2

(13)

Such an approximation not only has a modest computational cost, it is less prone to instabilities but also, as we will see later, produces more reliable results than AC-CAS. Here we apply the AC-CAS and AC0-CAS methods to predict excited-state energies of boron hydride, ethene, and a more challenging system such as s-tetrazine.

(10)

Eigenvectors normalized to −1/2 appear due to the existence of negative excitations (excitations from a considered state P to the lower-lying states) in the linear response function.26,27 5535


Letter

The Journal of Physical Chemistry Letters

states by the state-average procedure would yield stable solutions with constant numbers of negative excitations at α = 0 and perfectly smooth AC0 energy surfaces. The rotation of the methylene group around the CC bond in an ethylene molecule is a simple problem requiring a multireference wave function. While CAS(4e,4o) is sufficient to describe the static correlation effects (see Figure 2), without

The results for the boron hydride are presented in Figure 1. As a benchmark, full configuration interaction (FCI) data is

Figure 1. First three singlet excitation energies for a BH molecule along the BH bond length coordinate. CAS space employed was (6e,6o). Augmented cc-pVDZ basis set employed was taken from ref 29.

used, but we have also included CC3 results for comparison. The first excited singlet state, 11Π, of the BH molecule is correctly described even by the CASSCF(6e,6o) method. AC0-CAS and NEVPT2 methods are slightly more accurate than AC-CAS for short bond distances. The other two studied states, 21Σ+ and 11Δ, are more challenging. For 21Σ+ state at the equilibrium geometry, all of the multireference methods (bar the accidentally accurate CASSCF) yield an error of ca. 0.4 eV. Along the curve, however, AC-CAS, AC0-CAS, NEVPT2, and CC3 are in agreement. Again, the AC0-CAS slightly outperforms AC-CAS. Finally, the 11Δ state, which has a significant double-excitation contribution for short bond lengths29 and around RBH = 4 au changes its character to a single excitation due to the avoided crossing, is poorly described by the CCSD and CASSCF methods. AC0-CAS and AC-CAS are rather accurate at short interatomic distances, but for larger RBH, the error with respect to FCI values of both methods increases to almost 1.0 eV. For NEVPT2, the error after passing the avoided crossing region stays below 0.5 eV. Large errors observed for the 11Δ state are likely to result from the fact that CAS reference wave functions have been obtained in single-state calculations. The state-averaged CAS approach could result in smaller errors. A proper reference wave function for a given state should lead to a constant number of negative excitations in ERPA equations along the potential energy curve for the coupling parameters α = 0 (the reference wave function is an eigenfunction of the zeroth-order Hamiltonian) and α = 1. For a fixed geometry, it may happen that varying α from 0 to 1 leads to reordering of states and the number of negative excitations changes. For the CAS representation of the 21Σ+ excited state of BH, it has been observed that the number of negative excitations at α = 0 changes when RBH increases, and it also varies along the AC curve. It does not have a visible effect on the correlation energy, but upon closer inspection, we have found out that AC and AC0 (and also NEVPT2) potential energy curves show small discontinuities (of the order of less than 1 mHartree). One conjectures that improvement of the quality of the CAS description of excited

Figure 2. Relative (w.r.t. the ground-state minimum) energy of ethene molecule in the ground and the first excited state during twisting of the CC bond. The CAS space employed was (4e,4o) in the aug-cc-pVTZ30 basis set. Experimental values were taken from refs 31−33.

a dynamic correlation correction, CASSCF (especially the nonstate-averaged version used here) produces significant errors in excitation energies. AC, AC0, and PT2 successfully correct this behavior for the flat molecule (with AC-CAS being the closest to the experimental value). On the other hand, for the twisted geometry, NEVPT2 underestimates the excitation energy (see Table 1), while AC-CAS and AC0-CAS produce energies in Table 1. Vertical Excitation Energies for Flat (0°) and Twisted (90°) Forms of Ethylenea 0° 90°

Exp

CASSCF

AC0-CAS

AC-CAS

NEVPT2

7.8 2.7

7.0 3.4

8.2 2.6

7.9 2.9

8.2 2.2

a

Active space (4e,4o) (2 active orbitals of symmetry b2 and b3) and the aug-cc-pVTZ basis set were employed. The equilibrium geometry was taken from ref 34, and it was not optimized for other rotation angles.

good agreement with the experiment, the error for the two methods being 0.2 and −0.1 eV, respectively. Again it is worth noticing that the most efficient approach, AC0-CAS, produces also the most accurate result. Finally, let us look at the case of s-tetrazine, a challenging molecule with low-lying states of both valence and Rydberg character.35,36 A sensible active space to describe this molecule is (12e,10o). Such an active space has been employed in multistate CASPT2 based on a state-average CASSCF,34 which we use as a benchmark. Clearly (see Table 2), AC0-CAS results are comparable with those of NEVPT2 for the same active space. AC-CAS is shown to be a less reliable method 5536


Letter

The Journal of Physical Chemistry Letters Table 2. Excitation Energies of s-Tetrazine in eV Computed in the TZVP37 Basis Seta 1

1 B3u 11Au 11B1g 11B2u 11B2g 11B3g RMSE

CASPT2

CC3

CASb

AC0-CASb

AC-CASb

NEVPT2b

CASc

AC0-CASc

AC-CASc

2.24 3.48 4.73 4.91 5.18 5.79

2.53 3.79 4.97 5.12 5.34 − 0.25

3.69 5.18 6.09 7.61 6.31 7.93 1.83

2.30 3.52 5.26 5.47 5.47 6.26 0.39

2.77 4.07 5.68 6.17 5.79 6.92 0.89

2.29 3.40 5.05 5.39 5.18 5.84 0.24

3.50 4.77 6.27 5.09 6.13 7.46 1.25

2.21 3.50 4.89 5.25 5.09 5.77 0.16

2.72 3.85 5.39 5.34 5.46 6.32 0.47

a The geometry and CASPT2 (state-average, multistate variant) and CC3 results are taken from ref 34. Root-mean-square error (RMSE) computed with respect to CASPT2 reference. bThe CAS employed is CAS(12e,10o) as in ref 34. cThe CAS employed is CAS(14e,14o) as in ref 35.

house code, implemented as described in ref 20 and in the Supporting Information.

than AC0-CAS in this case. Notice that when one expands the active space from (12e,10o) to (14e,14o), the RMSE of AC0CAS goes down to 0.16 eV, while the cost of the computation of the AC0 correction is as low as that in the (12e,10o) space (see Table 3) and lower than the cost of computing the

■

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b02391. Details of the formula for implementation of the AC0 method (PDF)

Table 3. Timings of the Computation of the Correlation Corrections in the TZVP Basis Set for a Single Structure of s-Tetrazine in Seconds on a Single CPUa CAS

AC0

(12e,10o) (14e,14o)

3 × 10 3 × 101 1

ACb

NEVPT2

3 × 101 4 × 102

7 × 101 4 × 104

ASSOCIATED CONTENT

S Supporting Information *

■

a

AUTHOR INFORMATION

Corresponding Author

The NEVPT2 correction was computed in the Dalton software package and AC-CAS and AC0-CAS in our in-house code. bEstimate for a single grid point.

*E-mail: [email protected]. ORCID

Ewa Pastorczak: 0000-0002-5046-1476 NEVPT2 correction in the smaller space. The RMSE of the AC approach is lowered to 0.47 eV in the larger active space. The error is still above the one of the NEVPT2 method in a smaller active space, but it is worth noticing that AC-CAS is more efficient (by a 100 times) than the PT2 correction in the NEVPT2 method when the active space is expanded. In summary, we have shown first applications of the two methods based on the adiabatic connection formalism, AC and AC0, with a CAS reference function to predicting energies of excited states of molecules. Both methods rival approaches such as NEVPT2 and CASPT2 in accuracy. The proposed methods rely solely on one- and two-particle reduced density matrices of the reference function; therefore, they avoid a computational bottleneck of the perturbation methods, which is construction of higher-order RDMs. AC0, based on the linear approximation of the AC integrand, has turned out to be more reliable than AC. AC0 emerges as one of the most computationally efficient ab initio methods for CAS wave functions, the most expensive steps scaling as M2secM4act, MsecM5act, and M6act, where Msec and Mact denote the number of secondary and active orbitals, respectively. The presented applications included only singlet states, but the AC approaches can be extended to treat higher spin states and to employ state-average CASSCF reference wave functions.38 Work along these lines is in progress.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Center of Poland under Grant No. 2016/23/B/ST4/02848 (K.P.) and No. 2017/26/D/ST4/00780 (E.P.).

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REFERENCES

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COMPUTATIONAL DETAILS Throughout the Letter, the abbreviation NEVPT2 stands for a partially contracted variant of NEVPT2 and CCSD for equation-of-motion CCSD. Our CC3, CASSCF, and NEVPT2 results were obtained using Dalton software.39 All AC-CAS and AC0-CAS computations were performed using CASSCF reference wave functions from Dalton using our in5537


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Electronic Excited States from the Adiabatic-Connection Formalism

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