Four-component Polarization Propagator Calculations of Electron

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Four-component Polarization Propagator Calculations of Electron Excitations: Spectroscopic Implications of Spin-Orbit Coupling Effects Markus Pernpointner, Lucas Visscher, and Alexander B Trofimov J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01056 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 29, 2018

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Journal of Chemical Theory and Computation

Four-component Polarization Propagator Calculations of Electron Excitations: Spectroscopic Implications of Spin-Orbit Coupling Eects Markus Pernpointner,∗ † Lucas Visscher,‡ and Alexander B. Tromov¶ ,

E-mail: [email protected]

Abstract A complete implementation of the polarization propagator based on the DiracCoulomb Hamiltonian is presented and applied to excitation spectra of various systems. Hereby the eect of spin-orbit coupling on excitation energies and transition moments is investigated in detail. The individual perturbational contributions to the transition moments could now be separately analyzed for the rst time and show the relevance of one- and two-particle terms. In some systems dierent contributions to the transition moments partially cancel each other and do not allow for simple predictions. For the outer valence spectrum of the H2 Os(CO)4 complex a detailed nal state analysis is performed explaining the sensitivity of the excitation spectrum to spin-orbit eects. Finally, technical issues of handling double group symmetry in the relativistic framework and methodological aspects of our parallel implementation are discussed. ∗ † ‡

To whom correspondence should be addressed Theoretische Chemie, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Theoretical Chemistry, Faculty of Science, Vrije Universiteit Amsterdam, De Boelelaan 1083, NL-1081HV

Amsterdam, Netherlands

¶

Laboratory of Quantum Chemistry, Irkutsk State University, Karl Marx Str. 1, 664003 Irkutsk, Russia

1

ACS Paragon Plus Environment

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1

Introduction

Accurate theoretical predictions of absorption and emission spectra are of fundamental interest for the development of new optical devices and materials. A very important substance class possessing high quantum yields by triplet harvesting is based on metal organic complexes with heavy transition metals as central ion. For a theoretical description of the absorption and emission properties not only the wavelengths but also the corresponding transition moments need to be accurately calculated. We note thereby that for symmetric molecules moments are mostly determined by the transformation properties of the participating electronic wave functions in the point group of the molecule. To obtain accurate predictions a broad variety of ab initio methods can be used for small to medium-sized systems such as the approximate coupled cluster singles and doubles approach (CC2) 1,2 and its extension to triples (CC3) 36 within the linear response approach, 79 various models within the equationof-motion approach, 1012 multi-reference conguration expansion (MRCI) methods, 13 perturbational complete active space approaches, 14,15 multi-conguration self-consistent eld (MCSCF) treatments, 16 hybrid methods such as MRCI/DFT, 17 multireference coupled cluster (MRCC) approaches 1820 and polarization propagator methods using the algebraic-diagrammatic construction (ADC) 2131 or superoperator formalisms. 3234 The calculation of large systems is particularly prominent in current theoretical research and requires computation2


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ally less demanding methods such as conguration interaction singles (CIS), time-dependent Hartree-Fock (TDHF) and time-dependent density functional theory (TDDFT) that are applied with great success (see Ref. 35 for these approaches). For systems possessing a heavy central ion the accurate determination of excited states necessitates a balanced treatment of scalar relativistic, spin-orbit and electron correlation eects. It has turned out that relativity and electron correlation are in general not additive and correlated methods directly based on the relativistic Dirac formalism are therefore valuable. Not all of the above-mentioned methodologies, however, are available in combination with the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians. Since in presence of the DC or DCB Hamiltonians the orbitals are no longer separable into space and spin parts, excited state energy surfaces can not be characterized in terms of singlet or triplet character. As a consequence, calculation of intersystem crossing (ISC) rates employing Fermi's Golden Rule in combination with transition matrix elements over a spin-orbit Hamiltonian will not be applicable straightforwardly anymore. Instead, methods that are preferably used in vibronic coupling theory together with a diabatization of the accompanying states will play a major role. For a concise overview on the implications of spin-orbit coupling in photochemistry we recommend the reviews by C. M. Marian. 36,37 One of the most important distinctions of the DC Hamiltonian-based propagator approach

3



to established methods including spin-orbit coupling such as multireference spin-orbit conguration interaction (MRSOCI), 38 MRCI/DFT 17 or multi-state complete active space approaches in combination with perturbation theory (MS-CASPT2) 39 is that in the former one the correlation space is built from an orbital manifold fully relaxed with respect to scalar relativistic eects and spin-orbit coupling. In the latter approaches the correlation space is initially constructed from nonrelativistic or scalar relativistic conguration state functions and spin-orbit coupling is treated in a subsequent step. A relaxation of the one-particle functions upon sizeable spin-orbit coupling is therefore not taking place. For light to medium atoms a perturbational calculation of these eects is justied because the orbital relaxation upon spin-orbit coupling is not very large. This situation will change, however, when it comes to excitations that involve orbitals that undergo major changes upon inclusion of spin-orbit (SO) coupling and substantially participate in the excited states. In order to provide an ecient, size consistent method for the routine calculation of excitation spectra, the polarization propagator in a four-component framework has been recently realized 40 and for its numerical solution the Algebraic-Diagrammatic construction (ADC) 2123 was employed. In 40 we outline the relativistic extensions to the polarization propagator formalism with respect to excitation energy calculations in a double group framework. At this time of development transition moments were not yet available. Comparison to experiment,

4


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however, is only possible under inclusion of the moments associated with each individual excited state. This gap is now closed in this work and we will outline the eects of spin-orbit coupling on transition moments and outer valence absorption spectra for various systems. Larger systems can now be tackled by parallel implementations of the propagator and the Davidson diagonalizer.

2 2.1

Theory The Polarization Propagator

Since the essential theory for the relativistic polarization propagator was already outlined in earlier work 40 we can keep this section brief. The energy-dependent form of the polarization propagator reads as 41

Πpq,rs (ω) =

X hΨ0 |ˆ c†r cˆs |Ψ0 i c†q cˆp |Ψn ihΨn |ˆ n6=0

|

ω − (En − E0 ) + iη {z

X hΨ0 |ˆ c†q cˆp |Ψ0 i c†r cˆs |Ψn ihΨn |ˆ n6=0

}

Π+ (ω)

+

−ω − (En − E0 ) + iη

|

{z

Π− (ω)

.

(1)

}

In this expression Ψ0 and E0 represent the exact ground state wave function and energy and the creation (annihilation) operators cˆ†q (ˆ cp ) refer to the one-particle Hartree-Fock wave functions. Hereby, the positive innitesimal iη guarantees the convergence of the Fourier transformation between time and energy representations. The two parts of the polarization propagator Π+ (ω) and Π− (ω) are related to each other 21 and contain the same physical

5



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information. By applying diagrammatic techniques, a matrix representation of Eq. (1) can be derived whose diagonalization yields the transition energies and nal state eigenvectors. An alternative and possibly more comprehensive way to the derivation of the ADC matrix is given by the intermediate state representation (ISR). 4244 In general, one is interested in a systematic classication of an (exact) excited state of the system in terms of electronic congurations. This is straightforwardly realized by the well-known conguration interaction (CI) approaches. However, truncated CI methods exhibit some undesirable properties such as lack of size consistency and slow convergence. In the propagator/ISR approach the shifted

ˆ − E0 is now represented in a particular correlated and orthonormal Hamilton operator H ˜ I that is (formally) constructed by application of the physical excitation operators basis Ψ ˜ Ii CˆI to the exact ground state wave function |Ψ0 i. In this intermediate state basis set |Ψ the shifted Hamilton operator is represented as

˜ I |H ˆ − E0 |Ψ ˜ Ji MIJ = hΨ

(2)

ˆ is in our case with MIJ being identical to the diagrammatically derived ADC-matrix and H the Dirac-Coulomb operator

ˆ =H ˆ DC = H

N X

c~ αi · p~i + βi me c2 + Vext (i)14 +

i=1

N X 1 i b, i > j .

abij

∗ ∗ Here, Xn,ai and Xn,abij denote the ph and 2p2h components of the nth eigenvector X†n . In

our implementation we are able to calculate individual contributions to Tn that are grouped according to the perturbational orders of the F-vector and we observed a considerable difference in the various contributions (see Sec. 4.2). In particular, we split the resulting Tn according to

Tn =

X

n

(0)

(1)

(2),A,B,C

∗ Xn,ai Fai + Fai + Fai

(2),1

+ Fai

(2),10

+ . . . + Fai

ai

o

+

X

(2)

∗ Xn,abij Fabij .

(17)

abij

The 2p2h components of the F-vector only occur in second order. As mentioned above, an ADC eigenvector transforms according to a specic boson irrep Γµ (where the adjoint vector X†n then transforms according to Γ∗µ ) of the underlying molecular point group. In the contraction (11) we only obtain nonvanishing contributions to Tn if the F-vector symmetry is identical to Γµ leading to Γ∗µ ⊗ Γµ = Γ0 as required. From a technical point of view the contractions necessary for the F-vector calculation are achieved by resorting the integral streams for contiguous contraction ranges in memory. As soon as contractions over Dpq elements are to be performed (see eq. 12), nontotally symmetric

13



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extensions of the sorting routines have to be applied. This should briey be demonstrated (2)

for the Fabij vector components that read as

(2)

Fabij =

X k

+

X c

X hab||kii hab||kji Dkj − Dki + a + b − k − i k a + b − k − j

(18)

X hca||iji hcb||iji Dac − Dbc . c + b − i − j c c + a − i − j

For later use the rst part incorporating the occupied-occupied dipole moment integrals Dkj and Dki should be denoted as 2p2hA and the latter one involving the virtual-virtual dipole moment integrals Dac and Dbc as 2p2hB . Abbreviating the energy-containing fractions in the 2p2hA part as vab,ki and vab,kj and taking into account that vab,ki = −vab,ik we arrive at the expression (2)

Fabij =

X

vab,ki Dkj −

X

vab,kj Dki + . . .

(19)

k

k

In order to contract over k we resort −vab,ik to an intermediate array Aabi,k and contract with Dkj to another array Babi,j . Additionally, it can be seen that the expression above also leads to an i, j antisymmetrized F-component which necessitates a backsorting algorithm (2)

from Babi,j to Fabij . As a requirement for the contributing irreps we now have

γa∗ ⊗ γb∗ ⊗ γi ⊗ γj = Γµ

necessitating the extended logic in the sorters. Similar steps are to be performed for the 14


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2p2hB part.

2.3

Parallelization

In iterative diagonalization schemes where only a small number of eigenvalues are needed, the total Hamiltonian can be projected onto a smaller subspace which is exactly diagonalized and whose eigenvalues form increasingly precise approximations to the true eigenenergies. For our purpose we implemented the Liu modication 45 of the original Davidson algorithm. 46 For this algorithm, the matrix times trial vector multiplication is the most time consuming step and parallelization was achieved by slicing the large matrix into equal chunks for each individual compute node including the master node and forming partial matrix times vector products on each node. The sparsity of the Hamilton matrix is utilized in an index-driven storage scheme where the row and column index are stored together with a particular nonzero matrix element. This leads to a simple slicing of the matrix and the nal vector product can be constructed with the reduce/add operation of the message passing interface (MPI). After solving the subspace problem on the master node, new trial vectors are communicated to the compute nodes for the next iteration step. In Fig. 2 the parallelization scheme is illustrated focusing on the matrix vector multiplication. Hereby one should note that horizontal slices on the nodes N1 , . . . , Nn represent data chunks of equal size containing row and column indices together with the actual matrix entries. The resulting nonzero elements 15



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in the product vectors are therefore scattered over the whole vector length. We already work on more ecient storage schemes for this particular bottleneck of the algorithm. N1

N1

N2

N2

N3

N3

H X

(i)

Y

N n−1

(i)

N n−1

Nn

Nn

Compute Nodes

MPI broadcast

N1 N2 N3

MPI reduce/add

Y

(i) (i+1)

X N n−1 Nn

Master Node

Figure 2: (Color online). Implemented parallelization scheme for the partial matrix vector multiplication. The overall matrix H is partitioned on the nodes N1 , . . . Nn leading to the partial products Y (i) . These are collected and added on the master node to the complete vectors X (i+1) for the next iteration.

16


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3

Computational details

For argon we employed an uncontracted (18s11p9d) primitive basis previously used by Tromov and Schirmer. 22 The nonrelativistic Hartree-Fock ground state energy obtained in ref 22 amounts to -526.809304 Hartree, where our value of -526.808585 Hartree is slightly higher in energy due to the removal of totally symmetric contributions to the basis functions being equivalent to a transformation to spherical harmonics. Additionally, a point charge nucleus was applied for the nonrelativistic calculations whereas a Gaussian nuclear charge distribution is necessary for relativistic treatments in order to avoid singular behaviour of the wave function in close vicinity to the core. The use of uncontracted basis functions generally leads to very high-lying virtuals not being included in the active correlation space. For the strict ADC-2 calculation (ADC-2s) of the argon excitations and transition moments where the o-diagonal elements in the satellite block are zero we froze the 1s orbital and deleted virtuals above 500 a.u. (≈ 13600 eV) resulting in 174 active spinors. The geometry used for the hydrogen sulde calculations was taken from. 47 Hereby the HS distance amounts to 1.3356 Å and the H-S-H bond angle to 92.11 degrees. Due to the smallness of the system we could aord an aug-cc-pVTZ basis on each atom taken from 48,49 still leading to a moderately large active space of 142 spinors. 17



For the osmium complex H2 Os(CO)4 cc-pVTZ basis sets on each atom were employed and the geometry optimized within the density functional framework. We obtained a C2v structure whose xyz coordinates can be found in the appendix. This complex was chosen due to its compact ligand environment keeping the numerical eort at a minimum and, simultaneously, possessing a heavy metal center inferring sizeable spin-orbit coupling. The small component basis increases the total number of functions in the uncontracted calculation up to 2468 already asking for a parallel computation. As always, the X2Cmmf (innite-order twocomponent) transformation was applied prior to the AO/MO transformation step eliminating the necessity to generate half-transformed (SS|ik) integrals. 50 Due to the requirement using uncontracted basis sets for a good description of kinetic balance a dense manifold of molecular orbitals arises making the energy selection of the correlation space a nontrivial task. In order not to exceed the computational resources we restricted the active space to −0.7 . . . + 1.25 a.u. (−19 . . . + 34 eV) resulting in 194 spinors. Larger calculations approaching 300 active spinors are underway but they are not necessary for what we want to demonstrate in this work. Another important entity characterizing the type of transition is the pole strength (PS) formed as the sum of squares of the eigenvector components in the ph conguration space. In the case of excitations it reveals the amount of single excitation character of a particular

18


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nal state with a maximum value of 1.0. This, of course, is never achieved due to the presence of 2p2h congurations accounting for electron correlation. In the case of single and double ionizations calculated with propagator methods the pole strength is related to the observable peak height. For time measurements with a various number of cores we chose the HOSF molecule in C1 symmetry with a double-zeta basis on each atom. The absence of symmetry elements leads to complex two-electron integrals and the real and imaginary contributions to the transition moments can freely mix depending on initial conditions and orientation of the molecule. However, the squared sum of all contributions yields identical oscillator strengths as it must be the case. The active correlation space for HOSF comprised 110 spinors, leading to long enough integral batches for comparing run times on various numbers of cores.

4 4.1

Results and discussion Atomic excited states

In this subsection we will demonstrate the usefulness of the relativistic propagator for excited states calculations of the argon atom including spin-orbit coupling. According to the statements in sec. 2 the ADC matrix in nonrelativistic, spin-free (corresponds to scalar relativistic) or four-component representation is diagonalized per bosonic symmetry Γµ leading

19



to a set of corresponding excitation energies and eigenstates in the ISR basis. In a spinfree calculation the excited states are classied by their angular and spin quantum numbers

L and S , respectively, leading to an overall degeneracy of g = (2L + 1)(2S + 1). This degeneracy is reected by the occurrence of repeated eigenvalues in dierent Γµ that are automatically determined by the implementation. Then the total g can be determined by collecting all eigenvalues in the various Γµ which allows for a determination of L and S in the corresponding excited state. However, for some L, S combinations ambiguities can arise (see tab. 1) that can be resolved by application of Hund's rules. The angular and spin momenta are then further coupled to the total angular momentum J with allowed values of

J = |L + S|, |L + S − 1|, . . . , |L − S| being of dierent energy only if spin-orbit coupling is taken into account. Therefore, states of dierent J are not further discerned in the spin-free case. In order to demonstrate the applicability of the method we show nonrelativistic (tab. 1) and four-component (tab. 2) results for the lowest excitation manifold of argon. The type of excitation can be extracted from the corresponding eigenvector Xn in combination with a Mulliken population analysis. From the energies in tab. 1 one observes that the 3p5 4s1 excitation manifold is clearly separated from the 3p5 4p1 manifold where according to Hund's rules the 3 S term is located below the 1 P . Since excitation processes in atoms are calculated in lower symmetry than

20


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Table 1: Computed nonrelativistic excitation energies of argon in electronvolt using the ADC-2s method. The degeneracy g corresponds to the value (2L + 1)(2S + 1) and the 3p5 4p1 conguration exhibits two g = 3 degeneracies marked by an asterisk. All three Cartesian congurations 3p5x,y,z contribute to state 2. The abbreviations PS and OS stand for pole strength and oscillator strength, respectively. ST 1 2 3 4 5 6 7 8

conf. 3p5 4s1 3p5 4s1 3p5 4p1 3p5 4p1 3p5 4p1 3p5 4p1 3p5 4p1 3p5 4p1

calc Eexc. [eV] g 11.4078 9 11.5756 3 12.5994 3∗ 12.8356 15 12.8875 5 12.9585 3∗ 12.9588 9 13.0894 1

L 1 1 0 2 2 1 1 0

S 1 0 1 1 0 0 1 0

Term 3 P 1 P 3 S 3 D 1 D 1 P 3 P 1 S

PS 0.960 0.957 0.961 0.957 0.955 0.955 0.955 0.951

OS [au] 0.000 0.300 0.000 0.000 0.000 0.000 0.000 0.000

spherical, the transition moment for state 2 of 0.300 a.u. was obtained as the sum of three individual moments emerging from the 3p5x,y,z 4s1 excitations. Each of these transitions appears in a dierent boson symmetry belonging to the same 1 P term. Transition moments for 3p5 4p1 excitations vanish identically. The nonrelativistic results can not be directly compared to experiment where spin-orbit coupling is inherently present but they serve as an important reference for verifying symmetry relations and degeneracies. We now turn to the discussion of the four-component results given in tab. 2. As before, the overall degeneracies are determined by collecting eigenvalues from all possible nal states of symmetry Γµ and are now considerably reduced. For the 3p5 4s1 type of excitation we have two J = 3 cases separated by a gap of 0.2 eV. Eigenvector analysis reveals that the source orbitals for the corresponding states are 3p3/2 and 3p1/2 , respectively, where the j = l − s orbital participates in the state of higher excitation energy due to a larger stabilization of the 21



Page 22 of 46

Table 2: Four-component excitation energies and oscillator strengths of argon computed by using the ADC-2s method. For comparison with experiment 51 the values are also given in wavenumbers and the energy dierences between states are listed for an estimation of the quality. The abbreviations PS and OS stand for pole strength and oscillator strength, respectively. ST 1 2 3 4 5 6 7 8 9 10 11 12 13 14

conf.

3p53/2 4s1 3p53/2 4s1 3p51/2 4s1 3p51/2 4s1 3p53/2 4p1 3p53/2 4p1 3p53/2 4p1 3p53/2 4p1 3p53/2 4p1 3p53/2 4p1 3p51/2 4p1 3p51/2 4p1 3p51/2 4p1 3p51/2 4p1

calc Eexc

calc Eexc

−1

∆E calc

exp Eexc

−1

∆E exp

g

J

PS

OS [au]

5

2

0.960

0.0000

606.839

3

1

0.959

0.0234

803.068

1

0

0.959

0.0000

846.163

3

1

0.957

0.0775

3

1

0.961

0.0000

7

3

0.957

0.0000

154.511

5

2

0.957

0.0000

106087.305

469.990

3

1

0.956

0.0000

106237.597

150.292

5

2

0.956

0.0000

717.9

107054.319

816.722

1

0

0.954

0.0000

104634.1

224.8

107131.755

77.436

3

1

0.956

0.0000

104777.4

143.3

107289.747

157.992

5

2

0.956

0.0000

13.0172

104993.5

216.2

107496.463

206.716

3

1

0.955

0.0000

13.1234

105849.6

856.1

108722.668

1226.205

1

0

0.952

0.0000

[eV]

[cm

]

[cm

]

11.3156

91268.4

93143.800

11.3897

91866.2

597.7

93750.639

11.4965

92727.8

861.6

94553.707

11.5916

93494.5

766.7

95399.870

12.5637

101335.6

12.7576

102899.3

1563.7

105462.804

1360.660

12.7773

103058.3

159.0

105617.315

12.8389

103554.7

496.4

12.8558

103691.4

136.7

12.9448

104409.3

12.9727 12.9904

104102.144

3p1/2 in the atom. Despite some noticeable deviations in the onset of the excitation manifold the energy dierences between the states are in very good agreement with experiment for most of the states. There are states, however, that are not described to the same level of accuracy leading to larger deviations in the energy separations. A good agreement with experimental high-resolution measurements of oscillator strengths done by Chan et al. 52,53 could be achieved for the four-component calculations. Chan et al. obtained 0.0662 for the 3p53/2 4s1 and 0.265 for the 3p51/2 4s1 transitions located at excitation energies of 11.624 eV and 11.828 eV, respectively. For a comparison one needs to multiply our values of state 2 and 4 from table 2 by the degeneracy factors (3 in both cases) and one obtains 0.0701 and 0.2325 agreeing well with experiment. The sum of the latter two 22


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oscillator strengths yields the nonrelativistic value of state 2 in table 1 as it should be. Six substates contribute to the oscillator strengths of state 2 and 4 from which it becomes evident that some of the nonrelativistic 3 P components now participate in the spin-orbit coupled excitation manifold. Already for the lighter argon atom SO coupling introduces new transitions and leads to intensity redistributions. For the 3p5 4p1 excited state conguration considerable structure is added in the four-component case. All states derivable from the application of the jj coupling scheme for two independent p electrons are covered in the calculation albeit none of these states carries noticeable oscillator strength.

4.2

Transition moments

As mentioned above, transition moments are necessary to connect the ADC eigenvectors to the experimentally observable spectra. In the following we will report and compare the individual contributions to the TMs for the weakly and strongly relativistic systems H2 S and H2 Os(CO)4 , respectively. For transition metal complexes it is recommended to estimate the quality of a single reference description which is required for good propagator results (see 54 for a recent overview). A coupled cluster singles-double calculation of the osmium complex yielded a T1 diagnostics value of 0.039, still in the range for a valid single reference ground state description. 23



In Figs. 3 and 4 the individual TM contributions for both systems are shown where the zeroth order value serves as the reference due to its dominant contribution. For both systems the transition with the highest oscillator strength was chosen for the detailed analysis which is the rst singlet state in the case of H2 S and the fth state for the osmium complex, respectively. For H2 S we compare nonrelativistic and four-component results whereas for the heavy osmium system we intend to exclusively reveal the eect of spin-orbit coupling by comparing spin-free and four-component results. 0.08 ph 2,3

11 00 nonrelativistic 00 11 00 11 00 11 00 11 four−component 00 11 00 11

0.06

0.02

2p2h B

H 2S

0.04

ph 1 ph 2,1 ph 0

ph 2,7 ph 2,8

ph 2,5

ph 2A

2p2h A

Transition moment for strongest transition

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

Page 24 of 46

0 ph 2,6

ph 2C ph 2B

ph 2,2

ph 2,9 ph 2,10

-0.02

-0.04

-0.06 ph 2,4

-0.08 Contributions

Figure 3: (Color online). Individual contributions to the H2 S transition moment for the most intensive transition with the zeroth order contribution as a reference.

At rst it can be observed that in the weakly relativistic case of H2 S the nonrelativistic and 24


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four-component TM contributions to each perturbational order are nearly identical which does not come as a surprise but needs to be veried. Additionally, the ph 2, 3 and ph 2, 4 parts nearly cancel each other which will no longer occur in the osmium complex. Furthermore, the two-particle 2p2hB contributions should not be neglected even if a larger numerical eort is required for their calculation. The ndings applicable to H2 S change substantially in the osmium complex where sizeable deviations purely introduced by spin-orbit coupling are observed. As it was the case for H2 S, the 2p2hB part is still important in the osmium complex but now the 2p2hA elements start to gain relevance. Unlike for H2 S the ph 2, 3 and

ph 2, 4 parts no longer cancel but still are of opposite sign, a fact that makes predictions nearly impossible for general cases. Therefore the cancelation in H2 S can be considered to be accidental. In both molecules the 2p2hB contributions outweigh the 2p2hA ones explicable by the large number of V V -type dipole moment integrals and a correspondingly larger number of terms in the contractions. The nal oscillator strength of a particular transition calculates as

2 (x) 2 (y) 2 (z) 2 fn (E) = E Tn + Tn + Tn 3

(20)

where it is important to realize that in the low-symmetry cases the Tn(x,y,z) components take on complex values whose real and imaginary parts can freely mix under varying initial

25



0.06 ph 2,3

0.04

ph 2,9

ph 2,10

0.02 ph 2,2

ph 2C ph 2A

ph 2,5

ph 0

ph 2,1 ph 2B

ph 2,6 ph 2,4

2p2h B

ph 1

0

2p2h A

Transition moment for strongest transition

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

Page 26 of 46

-0.02 ph 2,7

ph 2,8

-0.04

11 00 00 spin−free 11 00 11 00 11 00 11 four−component 00 11 00 11

-0.06

-0.08

H 2 Os(CO) 4

-0.1 Contributions

Figure 4: (Color online). Individual contributions to the H2 Os(CO)4 transition moment for the most intensive transition with the zeroth order contribution as a reference. conditions. The resulting oscillator strengths (20), however, are always independent of initial conditions such as the reference orientation of the molecule. In the following we analyze the eect of spin-orbit coupling on the outer valence excitation spectrum of H2 Os(CO)4 with its heavy central metal. In g. 5 the stick spectra obtained by a spin-free and four-component ADC-2s calculation are shown where the spin-free spectrum is plotted downwards. The peak heights reect the amount of single excitation character of a particular transition calculated by squaring the coecients of the ADC eigenvector within the ph manifold. In case of a sizeable double excitation contribution an extended 26


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ADC treatment including the o-diagonal terms in the satellite block may be in order for a better description of the corresponding nal state. In our case, all peaks within the outer valence energy range have negligible 2p2h character and an extended treatment also being computationally more demanding is not required. 1.5

four−component 1

0.5

0

-0.5

-1

spin−free singlet spin−free triplet -1.5 5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

Figure 5: (Color online). Four-component (upwards) and spin-free (downwards) excitation stick spectrum of H2 Os(CO)4 in the outer valence region. The peak height corresponds to the fraction of single excitations forming the state. It can be seen that inclusion of spin-orbit coupling leads to substantial energetic shifts and an increased number of peaks.

In the next step we take the computed transition moments into account and arrive at a more physical picture of the spectrum to be expected in experiment. Due to spin selection rules in nonrelativistic theory all the triplet peaks in the spin-free spectrum disappear and the remaining singlet peak heights are now proportional to the corresponding oscillator strengths 27



from (20). For taking experimental resolution into account we convolute the resulting stick spectrum with a Lorentzian envelope of 30 meV width at half maximum and arrive at Fig. 6 where the nal spectrum is shown.

four−component spectrum (upwards)

Intensity

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

Page 28 of 46

spin−free spectrum (downwards)

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

Excitation energy [eV]

Figure 6: (Color online). Four-component (upwards) and spin-free (downwards) excitation spectrum of H2 Os(CO)4 convoluted with a Lorentzian envelope of 30 meV width at half maximum. The individual peak heights correspond to the calculated transition moments. The underlying stick spectrum is shown in green color.

Inclusion of SO-coupling induces additional structure in the low-energy range of the excitation spectrum in combination with a redistribution of intensity. Additionally, the two major features in the spectrum undergo a red shift by approx. 0.1 eV and the number of states is markedly increased. For the outer-valence spectrum one expects dominant transitions from the occupied CO π orbitals to the corresponding π ∗ orbitals. Therefore the question arises

28


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how spin-orbit coupling can lead to the observed structural changes. This is only possible if metal orbitals contribute notably to the excited state congurations. To analyze this further we performed the kind of state population analysis mentioned at the beginning of the text. In general, a characterization of an excited state conguration is obtained by looking at the occupied and virtual spinors in the leading determinant of the correlation expansion. As we saw before, all outer valence nal states exhibit over 90% single excitation character and the corresponding expansion coecient of the leading determinant is also above 0.9. It is therefore a valid approach to look at the atomic compositions of the molecular source and destination spinors forming the dominant hole/particle excitation in each state. If we pursue such a procedure we arrive at g. (7) where the atomic hole (particle) populations of each nal state are shown in the upper (lower) half of the picture. For the rst group of excited states the hole space shows dominant carbon s/p, oxygen p and hydrogen s composition with noticeable osmium contributions. This changes for the subsequent group of states where sizeable metal-ligand charge transfer takes place (5.8 6eV). The presence of large osmium contributions in each state rationalizes the considerable inuence of spin-orbit coupling on the spectrum. Due to the relatively low symmetry of the molecule considerable mixing of atomic spinors at dierent centers can occur as can be seen in the gure. The considerable hydrogen s portion in the hole space arises from

29



a participation of the bonding orbital between hydrogen and an occupied osmium d spinor serving as source orbitals for excitation. In the particle space the accepting orbitals exhibit large carbon p and oxygen p contributions pointing to a π ∗ character. However, unoccupied osmium p orbitals are also mixed in and can act as acceptors for electron density in the excited state. Spin-orbit splitting in the osmium virtual p orbitals is considerably less pronounced than in the occupied d spinors indicating that SO-induced shifts play a larger role in the hole space of the nal state than in the particle space. In summary, both the hole and particle space possess pronounced metal character and therefore react sensitively to changes in the osmium spinors that occur upon inclusion of spin-orbit coupling.

4.3

Parallelization

For large problems with ADC matrix (H) dimensions exceeding one million or more the

|cj i = H|bj i multiplication becomes the most time-consuming step in the algorithm. The vectors |cj i are needed for the projection of H on the trial vector subspace formed by the |bj i

˜ ij = hbi |H|bj i = hbi |cj i and for the calculation of the residual vectors |rj i according to (H) for each one of the sought eigenstates. If memory resources allow for it, internal storage of the vectors |cj i, j = 1, . . . , m is optimal, while otherwise we use external storage. We tested both alternatives and found that all of them are faster than recalculating the projection 30


Page 30 of 46

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˜ ij . Considering the construction of the projection (H) ˜ ij numerical eort is minimized by (H) ˜ ij following the scheme depicted in exclusively calculating the new blocks in the matrix (H) g. 8. Converged roots do not induce an extension of the trial vector space which leads to considerably shorter computing times for the projection at the end of the Davidson run. The example in the gure is drawn for four microiterations where the subspace grows by nr in each step. Should the desired convergence of the roots not have been reached, we reenter the algorithm with a new set of nr trial start vectors |˜bj i now constructed from the available space according to

|˜bj i =

N X

akj |bk i ,

j = 1, . . . , nr

k=1

˜ ij where the akj are the eigenvector components of the (exactly) diagonalized projection (H) possessing dimension N . Once this new set of start vectors is generated we build up a new trial space corresponding to the upper left part in Fig. 8. In actual implementations this procedure is often termed as collapsing the subspace and we follow this convention here. In subsection 2.3 we outlined the parallelization strategy by distributing equally sized chunks of the matrix over the nodes. In table 3 we list the corresponding timings for the calculation of the HOSF excitation spectrum using one to eight cores. Hereby the distributed matrix multiplication was timed separately together with the overall time spent in the Davidson algorithm. It is observed that the crucial part forming H|bj i scales very well with increasing

31



Page 32 of 46

number of cores, although it is not perfectly linear. This is due to the fact that the completion of the partial vectors together with the forming of the scalar product hbi |cj i is done on the master only. The overall run times in Davidson do not show the same scaling eciency because all computations where the complete vectors are required have to be done on the master node and therefore introduce some amount of sequential execution. In all parallel runs the local disk storage requirements for the ADC matrix are reduced by a factor of 1/M where M is the number of cores. For considerably larger problems where the H|bj i step and matrix storage become serious bottlenecks the parallel version reveals its true value. Table 3: Measured compute times (in sec) for the matrix times vector part and the complete Davidson run. For larger systems the Krylov projection will be the most time consuming step and that is the place where the scaling is optimal. Number of cores Time for Krylov projection Total time in Davidson

5

1 290 12371

2 150 7980

4 90 7100

8 50 5200

Conclusions

In this work we presented a detailed analysis of the eect of spin-orbit coupling on excitation energies, transition moments and spectral structure for various systems together with an outline of the corresponding implementation available in the Dirac program package. 55 In the relativistic case the mathematical and computational frame have to be extended due to the occurrence of fermion irreps. This introduces complex molecular two-electron integrals 32


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also leading to complex transition moments and a substantially increased numerical eort. It could be demonstrated for H2 Os(CO)4 that SO eects alone introduce major changes in the expected absorption spectrum which will similarly hold for other metal complexes of the 5d series as, for example, iridium used in OLEDs. The polarization propagator in a four-component framework together with a suitable parallelization strategy provides a useful method to obtain excitation spectra of large systems with heavy centers where a perturbational treatment of SO eects may fail and where electron correlation and SO coupling need to be treated on an equal footing.

6

Acknowledgments

One of the authors (MP) gratefully acknowledges nancial support by the Deutsche Forschungsgemeinschaft.

33



0.8

Cd Cp Cs Hs Op Os d Os p

0.6

hole space

Composition

1

0.4

0.2

0 5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

Energy [eV]

Cp Cs Op Os d Os f Os p Os s

1

0.8

Composition

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

Page 34 of 46

particle space

0.6

0.4

0.2

0 5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

Energy [eV]

Figure 7: (Color online). Excited state compositions in terms of atomic contributions separated into occupied (hole) and virtual (particle) domains. Contributions from specic atomic s, p, d, . . . functions are hereby combined and depicted in a single color. For each excited state the upper part hereby shows the composition of the source orbitals the electron is excited from and the lower part does the same for the target orbitals accepting the electron.

34


Page 35 of 46 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


00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 n r

nr

m=1

m=2

00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 nr

m=3

m=4

00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

Collapsing the subspace

˜ ij = hbi |H|bj i for the Figure 8: (Color online). Construction of the subspace projection (H) microiterations m = 1, . . . , 4. In each microiteration the dimension of the trial space grows by nr at most. The new portions to be calculated are shown in full color, the reused parts in shaded color. The resulting projected matrix is Hermitian.

35



APPENDIX The x, y, z geometry of the osmium complex in C2v structure obtained at the DFT level is listed in the following table. Table 4: Cartesian coordinates of H2 Os(CO)4 in Å. Atom Os O O O O C C C C H H

x 0.00000 2.41214 -2.41214 0.00000 0.00000 0.00000 0.00000 1.51252 -1.51252 1.10331 -1.10331

y 0.00000 0.00000 0.00000 -3.03655 3.03655 -1.93462 1.93462 0.00000 0.00000 0.00000 0.00000

36

z 0.00000 -1.98152 -1.98152 0.59571 0.59571 0.31819 0.31819 -1.28285 -1.28285 1.25671 1.25671


Page 36 of 46

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(55) DIRAC, a relativistic ab initio electronic structure program, Release DIRAC17 (2017), written by L. Visscher, H. J. Aa. Jensen, R. Bast, and T. Saue, with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faÿhauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, E. D. Hedegård, T. Helgaker, J. Henriksson, M. Ilia², Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, J. K. Lærdahl, C. V. Larsen, Y. S. Lee, H. S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R. di Remigio, K. Ruud, P. Saªek, B. Schimmelpfennig, A. Shee, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, S. Villaume, O. Visser, T. Winther, and S. Yamamoto (see

http://www.diracprogram.org).

45



four−component spectrum (upwards)

Relative Intensity

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 5.4 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 46

spin−free spectrum (downwards)

5.6

5.8

6

6.2

Excitation energy [eV]


6.4

6.6

6.8

Four-component Polarization Propagator Calculations of Electron

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