Electronic Excited States from the Adiabatic-Connection Formalism

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Spectroscopy and Photochemistry; General Theory

Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wavefunctions Ewa Pastorczak, and Katarzyna Pernal J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02391 • Publication Date (Web): 07 Sep 2018 Downloaded from http://pubs.acs.org on September 8, 2018

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The Journal of Physical Chemistry Letters

Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wavefunctions Ewa Pastorczak and Katarzyna Pernal∗ Institute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924 Lodz, Poland E-mail: [email protected]

Abstract It is demonstrated how the recently proposed multireference adiabatic connection (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 excited states energies of molecules using a complete active space (CAS) wavefunction 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 1- and 2-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 of the accuracy rivaling that of the NEVPT2 method but at its marginal computational cost.

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ACS Paragon Plus Environment


TOC entry

E x c ite d S ta te E n e r g y

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A C 0 -C A S (1 -R D M , 2 -R D M ) N E V P T 2 (3 -R D M , 4 -R D M ) F C I ~ N !

B o n d L e n g th

Unlike the ground states, for whose qualitative description a single-determinant wavefunction 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 wavefunctions. 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-consistency 9 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 wavefunction. The need to calculate the 3- and 4-particle reduced density matrices 11 (RDMs) causes the cost of computation to grow as the 8th power of the number of active orbitals in CAS. Circumventing this requirement either through the use of cumulant approximations 12,13 or linear-response theory 14,15 has proven challenging. Alternatives to perturbation corrections include density functional theory approaches, 16,17 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, non-empirical approaches to calculating excited-state energies using a CAS reference. It is shown that, while considerably more efficient in treating ac∗

To whom correspondence should be addressed

2


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tive orbitals than perturbation approaches, the proposed AC-CAS and AC0-CAS methods produce results on par with such sophisticated methods as NEVPT2 and CASPT2. Assume that a given reference state Ψref is an approximate representation of the P th exact state. The error in the energy with respect to the exact value EP arising from using an approximate wavefunction will be called a correlation energy for the P th state D E ˆ ref Ecorr = EP − Ψref |H|Ψ

,

(1)

ˆ is a nonrelativistic electronic Hamiltonian of a given system. In Refs. 19 and 20 we where H have proposed the adiabatic-connection (AC) correlation energy formula for ground states. ˆ =H ˆ (0) + H ˆ 0 , where It has been based on a group-product partitioning of the Hamiltonian H ˆ (0) includes two-particle interaction operator in the active space, 21 and on defining the AC H ˆα = H ˆ (0) + αH ˆ 0 , with the coupling parameter α ranging from 0 to 1. The Hamiltonian as H AC formula for the correlation energy has been expressed in terms of transition RDMs γ α,ν , occupation numbers np and two-electron integrals hrs|pqi in the following form ˆ AC Ecorr

1

W α dα ,

=

(2)

0

where the AC integrand reads 1 X0 Wα = 2 pqrs

! X

α,ν α,ν γpr γsq + (np − 1)nq δrq δps

hrs|pqi

(3)

ν

and a prime indicates exclusion of terms involving all four indices pqrs belonging to either a set of active or inactive orbitals. The AC expression has been obtained under an assumption that a one-electron RDM stays constant along the AC path. Its extension to excited sates is straightforward, namely, if the reference wavefunction represents a P th exact state then transition RDMs connect the P th and νth states of the AC Hamiltonian, i.e.

α,ν ˆ α Ψα = E α Ψα . In the approximate AC formalism transition γpq = ΨαP |ˆ a†q a ˆp |Ψαν , where H ν ν ν

3


The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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RDMs are obtained as 22

(γ α,ν )qp =

   (np − nq ) [Yνα ]pq

∀p>q ,

  (np − nq ) [Xαν ]qp

∀q>p ,

(4)

where the vectors [Xαν , Yνα ] follow from the Extended Random Phase Approximation (ERPA) equations. ERPA provides extension of RPA for multireference wavefunctions 23,24 and it takes form of a generalized eigenproblem reading 











 A B   Xν   −N 0   Xν     = ων    B A Yν 0 N Yν

.

(5)

The matrices A and B are determined by the Hamiltonian and the reference wavefunction for a given system as follows

∀p>q

r>s

∀p>q

r>s

D h i E ˆ a Apq,rs = Ψref | a ˆ†p a ˆq , [H, ˆ†s a ˆr ] |Ψref D h i E ˆ a Bpq,rs = Ψref | a ˆ†p a ˆq , [H, ˆ†r a ˆs ] |Ψref

(6) (7)

and the metric matrix is given by natural occupation numbers np (pertaining to Ψref )

∀p>q

r>s

Npq,rs = δpr δqs (nq − np ) .

(8)

Throughout the paper indices pqrs correspond to natural spinorbitals 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, a ˆ†p a ˆq where 0 < np , nq < 1, are non-redundant. 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 activesecondary pq pairs in a ˆ†p a ˆq operators. Solutions of Eq.(5) come in pairs of positive and 4


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negative eigenvalues, ων+ and ων− , equal up to the absolute value, ων+ = −ων− , whereas the corresponding eigenvectors posses 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 normalization 20 T T 1 Yν+ N Yν+ − Xν+ N Xν+ = 2

.

(9)

In the case of Ψref representing an excited state one expects that in addition to eigenvectors normalized as shown in Eq.(9) there may exist vectors conforming to negative normalization

Yν+

T

T 1 N Yν+ − Xν+ N Xν+ = − 2

.

(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 When exploring potential energy surface of a given state, a proper reference wavefunction 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 wavefunction the number of negative excitations should be equal to P − 1, 28 however due to the approximate nature of the ERPA polarization propagator, frequently fewer of them can be observed. It is important to notice that for an excited-state-representing wavefunction the Hessian matrix is not positive-definite and the A−B and A+B matrices in general may have negative eigenvalues. 28 Consequently, the ERPA problem 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 real-value nonsymmetric eigenproblem 5



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reading

N −1 (A + B)N −1 (A − B) (Yν − Xν ) = ων2 (Yν − Xν )

(11)

1 (Yν + Xν ) = p N −1 (A − B)(Yν − Xν ) . ων2

(12)

and

For a stable reference Ψref the eigenvalues {ων2 } are real and positive. ERPA eigenvectors [Xαν , Yνα ] obtained from Eqs.(11)-(12) correspond to positive values of ων and they should be normalized as shown in Eqs.(9) and (10). Presence of the negatively normalized eigenvectors (negative excitations) is related to de-excitation modes in the ERPA propagator so it is only expected if the reference wavefunction describes an excited state. Employing the ERPA approximation for the reduced density matrices in the AC formalism allows one to obtain the correlation energy solely by means of one- and two-electron RDMs of the given wavefunction 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 it has been ˆ α=1 is expected to yield negativediscussed, ERPA corresponding to a full Hamiltonian H excitation-solutions, defined by Eq.(10), if Ψref represents an excited state. The same is true ˆ α=0 , since by construction it is assumed that in the non-correlated limit corresponding to H ˆ (0) = H ˆ α=0 Hamiltonian. Based on the observation that the AC Ψref is the P th state of the H integral is typically nearly-linear 19,20 the correlation energy has been further approximated α using the first-order expansion at α = 0, i.e. W α = W α=0 + dW α = W (0) + W (1) α, dα α=0 and noticing that W (0) = 0, which, after exploiting the formula in Eq.(2), has resulted in the AC0 approximation (see Supporting Information for details)

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 6


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hydride, ethene, and of a more challenging system such as s-tetrazine. The results for the boron hydride are presented in Fig. 1. As a benchmark FCI data is used but we have also included CC3 results for comparison. The first excited singlet state, 11 Π, of the BH molecule is correctly described even by the CASSCF(6e, 6o) method, but AC0-CAS and NEVPT2 methods are slightly more accurate than AC-CAS for short bond distances. The other two studied states, 2 1 Σ+ and 11 ∆ are more challenging. For 2 1 Σ+ state at the equilibrium geometry all the multireference methods (bar the accidentally accurate CASSCF) yield an error of c.a. 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 11 ∆ state, which has a significant double-excitation contribution for short bond lengths 29 and around RBH = 4 a.u. 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 11 ∆ state are likely to result from the fact that CAS reference wavefunctions have been obtained in single-state calculations. State-averaged CAS approach could result in smaller errors. A proper reference wavefunction 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 wavefunction 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 2 1 Σ+ 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, AC0 (and also NEVPT2) potential energy curves show small discontinu-

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ities (of the order less than 1 milihartree). One conjectures that improvement of the quality of the CAS description of excited 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.

3.5

7.0

8.0

CCSD CC3

3.0

6.5

CASSCF AC-CAS

Excitation energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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7.5

AC0-CAS

2.5

6.0

NEVPT2 FCI

2.0

5.5

1.5

5.0

1.0

4.5

7.0

6.5

0.5

1

4.0

1

2

0.0

1

1

6.0

1

3.5 2

3

4

5

6

R (a.u.)

7

8

2

3

4

5

6

7

R (a.u.)

8

2

3

4

5

6

7

8

R (a.u.)

Figure 1: First three singlet excitation energies for a BH molecule along the BH bond length coordinate. CAS space employed was (6e, 6o). Full configuration interaction (FCI) values and the augmented cc-pVDZ basis set employed were taken from Ref. 29. The rotation of the methylene group around the C=C bond in an ethylene molecule is a simple problem requiring a multireference wavefunction. While CAS(4e, 4o) is sufficient to describe the static correlation effects (see Fig. 2), without a dynamic correlation correction CASSCF (especially the non-state-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, 8


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for the twisted geometry NEVPT2 underestimates the excitation energy (see Table 1), while AC-CAS and AC0-CAS produce energies in a good agreement with the experiment, the error for both methods being 0.2 eV and −0.1 eV, respectively. Again it is worth noticing that the most efficient approach, AC0-CAS, produces also the most accurate result.

8

7

Relative energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6

CASSCF

5

AC-CAS AC0-CAS

4

NEVPT2 3

Experiment

2

1

0 0

15

30

45

60

75

90

Torsional angle (deg)

Figure 2: Relative (w.r.t. the ground state minimum) energy of ethene molecule in the ground and the first excited state during the twisting of the C=C bond. CAS space employed was (4e, 4o) in aug-cc-pVTZ 30 basis set. Experimental values taken from Refs. 31–33. Table 1: Vertical excitation energies for flat (0◦ ) and twisted (90◦ ) form of ethylene. Active space (4e, 4o) (2 active orbitals of symmetry b2 and b3 ) and aug-cc-pVTZ basis set were employed. Equilibrium geometry was taken from Ref. 34 and it was not optimized for other rotation angles. ◦

0 90◦

Exp CASSCF AC0-CAS AC-CAS NEVPT2 7.8 7.0 8.2 7.9 8.2 2.7 3.4 2.6 2.9 2.2

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 active space has been employed in multistate CASPT2 based on a stateaverage CASSCF, 34 which we use as a benchmark. Clearly (see Table 2), AC0-CAS results 9



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are comparable with NEVPT2’s for the same active space. AC-CAS is shown to be a less reliable method than AC0-CAS in this case. Notice that when one expands the active space from (12e, 10o) to (14e, 14o), the RMSE of AC0-CAS 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 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 NEVPT2 method when the active space is expanded. Table 2: Excitation energies of s-tetrazine in eV computed in TZVP 37 basis set. The geometry and CASPT2 (state-average, multistate variant) and CC3 results are taken from Ref. 34. CAS employed are a CAS(12e, 10o) as in Ref. 34 and b CAS(14e, 14o) as in Ref. 35. Root-mean-square error (RMSE) computed with respect to CASPT2 reference. 11 B

3u

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

CASPT2 2.24 3.48 4.73 4.91 5.18 5.79

CC3 2.53 3.79 4.97 5.12 5.34 0.25

CASa 3.69 5.18 6.09 7.61 6.31 7.93 1.83

AC0-CASa 2.30 3.52 5.26 5.47 5.47 6.26 0.39

AC-CAS 2.77 4.07 5.68 6.17 5.79 6.92 0.89

a

NEVPT2a 2.29 3.40 5.05 5.39 5.18 5.84 0.24

CASb 3.50 4.77 6.27 5.09 6.13 7.46 1.25

AC0-CASb 2.21 3.50 4.89 5.25 5.09 5.77 0.16

AC-CASb 2.72 3.85 5.39 5.34 5.46 6.32 0.47

Table 3: Timings of the computation of the correlation corrections in TZVP basis set for a single structure of s-tetrazine in seconds on a single CPU. NEVPT2 correction was computed in Dalton software package and AC-CAS and AC0-CAS in our in-house code. ∗ Estimate for a single grid point. AC∗ 3 · 101 4 · 102

CAS AC0 (12e, 10o) 3 · 101 (14e, 14o) 3 · 101

NEVPT2 7 · 101 4 · 104

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 10


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accuracy. The proposed methods rely solely on 1- and 2-particle reduced density matrices of the reference function so they avoid a computational bottleneck of the perturbation methods which is construction of higher-order RDM’s. 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 method 2 4 5 6 for CAS wavefunctions, the most expensive steps scaling as Msec Mact , Msec Mact , and Mact ,

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 wavefunctions. 38 Work along these lines is in progress.

Computational Details Throughout the paper 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 wavefunctions from Dalton using our in-house code, implemented as described in Ref. 20.

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

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(27) Olsen, J.; Yeager, D. L.; Jørgensen, P. Optimization and Characterization of a Multiconfigurational Self-Consistent Field (MCSCF) State. Advan. Chem. Phys. 1983, 54, 1. (28) Golab, J. T.; Yeager, D. L.; Jørgensen, P. Proper Characterization of MCSCF Stationary Points. Int. J. Quantum Chem. 1983, 24, 645–645. (29) Koch, H.; Christiansen, O.; Jørgensen, P.; Olsen, J. Excitation Energies of BH, CH2 and Ne in Full Configuration Interaction and the Hierarchy CCS, CC2, CCSD and CC3 of Coupled Cluster Models. Chem. Phys. Lett. 1995, 244, 75–82. (30) Dunning Jr., T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007. (31) Douglas, J. E.; Rabinovitch, B. S.; Looney, F. S. Kinetics of the Thermal Cis-Trans Isomerization of Dideuteroethylene. J. Chem. Phys. 1955, 23, 315–323. (32) Foo, P. D.; Innes, K. K. New Experimental Tests of Existing Interpretations of Electronic Transitions of Ethylene. J. Chem. Phys. 1974, 60, 4582–4589. (33) Grimme, S.; Waletzke, M. Multi-reference Moeller-Plesset theory: Computational Strategies for Large Molecules. Phys. Chem. Chem. Phys. 2000, 2, 2075–2081. (34) Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks for Electronically Excited States: Time-Dependent Density Functional Theory and Density Functional Theory Based Multireference Configuration Interaction. J. Chem. Phys. 2008, 129, 104103. (35) Schutz, M.; Hutter, J.; Luthi, H. P. The Molecular and Electronic Structure of stetrazine in the Ground and First Excited State: A Theoretical Investigation. J. Chem. Phys. 1995, 103, 7048–7057.

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(36) Angeli, C.; Cimiraglia, R.; Cestari, M. A Multireference n-electron Valence State Perturbation Theory Study of the Electronic Spectrum of s-tetrazine. Theor. Chem. Acc. 2009, 123, 287–298. (37) Schäfer, A.; Huber, C.; Ahlrichs, R. Fully Optimized Contracted Gaussian Basis Sets of Triple Zeta Valence Quality for Atoms Li to Kr. J. Chem. Phys. 1994, 100, 5829–5835. (38) Werner, H.-J.; Meyer, W. A Quadratically Convergent MCSCF Method for the Simultaneous Optimization of Several States. J. Chem. Phys. 1981, 74, 5794–5801. (39) Aidas, K. et al. The Dalton Quantum Chemistry Program System. WIREs Comput. Mol. Sci. 2014, 4, 269–284.

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Excitation energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44


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Electronic Excited States from the Adiabatic-Connection Formalism

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