Transition-dipole moments for electronic excitations in strong magnetic

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Quantum Electronic Structure

Transition-dipole moments for electronic excitations in strong magnetic fields using equation-of-motion and linear response coupled-cluster theory Florian Hampe, and Stella Stopkowicz J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00242 • Publication Date (Web): 29 May 2019 Downloaded from http://pubs.acs.org on June 10, 2019

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Journal of Chemical Theory and Computation

Transition-dipole moments for electronic excitations in strong magnetic fields using equation-of-motion and linear response coupled-cluster theory Florian Hampe∗ and Stella Stopkowicz∗ Institut f¨ ur Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany E-mail: [email protected]; [email protected]

May 28, 2019 Abstract An implementation of transition-dipole moments at the equation-of-motion coupledcluster singles-doubles (EOM-CCSD) and CCSD linear response (LR) levels of theory for the treatment of atoms and molecules in strong magnetic fields is presented. The presence of a finite magnetic field leads, in general, to a complex wave function and a gauge-origin dependence, necessitating a complex computer code together with the use of gauge-including atomic orbitals. As in the field-free case, for EOM-CC, the evaluation of transition-dipole moments consists in setting up the one-electron transitiondensity matrix (TDM) which is then contracted with dipole-moment integrals. In the case of CC-LR, the evaluation proceeds with a modified TDM but additionally requires a second contribution accounting for the amplitude response which is missing in EOM-CC theory for properties. We present a selected set of transitions for the sodium

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atom and investigate the LiH molecule in both a parallel as well as a perpendicular magnetic field. The dependence of excited-state energies and transition moments on the magnetic-field strength is discussed with a focus on magnetic-field induced avoided crossings. Additionally, the differences between field-dependent EOM-CCSD and CCSD-LR transition moments are investigated.

Introduction In magnetic fields with field strengths of around 1 a.u. (1a.u. = 1B0 ≈ 235000 T), Coulomb and magnetic forces in atoms and molecules compete. 1 Such fields can be found, e.g., on magnetic white dwarf stars 1–3 whose observable spectra therefore differ drastically from those of non-magnetic white dwarfs. In astrophysics, these spectra are used to obtain information about the field strength as well as the atmosphere’s chemical composition. 4 Undoubtedly, such high field strengths cannot be produced in laboratories on Earth. As such, only theory can provide reliable data to investigate the drastically altered electronic structure which is caused by strong magnetic fields. For weaker magnetic fields up to around 2000-5000 T, electronic transitions may be calculated using perturbation theory (Zeeman and Paschen-Back). 4 Beyond this limit, when the magnetic field is no longer a small perturbation, nonperturbative theories are mandatory. For the hydrogen atom, finite-field calculations have been performed (see, for example, Refs. 5–8) which have enabled the assignment of spectra from strongly-magnetized white dwarfs with hydrogen atmospheres. 9,10 Later, extensive calculations have also been carried out by Becken et al. for the helium atom at the full configuration interaction (FCI) level of theory 11–13 in order to determine the transition energies in strong magnetic fields. These calculations led to the discovery of the first strongly-magnetized white dwarf with a helium atmosphere and a field of 30-70 kT. 14–16 Due to the limited applicability of FCI calculations, corresponding predictions are only feasible for systems with a few electrons, like lithium, 17–19 beryllium, 20,21 or the hydrogen molecule. 22–24 However, it has already been shown that other 2


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elements (C, Na, Ca, Mg, Al, Fe, . . . ) 25–27 and even molecules with more than two electrons (CH, C2 ) 2,28 are present on weakly magnetized white dwarfs, suggesting that they may exist on strongly-magnetized white dwarfs as well. However, for their detection, theoretical predictions from finite-field calculations are crucial. Atoms with more electrons have already been studied at the Hartree-Fock (HF) level of theory in strong magnetic fields, e.g., boron, 29 carbon, 30 and sodium. 31 While such investigations provide a first insight into the behavior of atoms and molecules in strong fields, they are by far not accurate enough to be used to guide the assignment of observational spectra from highly magnetized white dwarf stars. In strong magnetic fields, electronic states evolve in a very complicated manner, leading to a plethora of state crossings and avoided crossings. Consideration of electron correlation has already proven to be of great importance in this context: as an example, for carbon, electron correlation shifts the calculated field strengths at which the ground state changes from 3 Π−1 to 5 Σ (3 P and 5 S in field-free notation) by several thousand tesla. 32 Consequently, for meaningful theoretical predictions of spectra, consideration of electron correlation is indispensable. In order to address this problem, we have previously presented a finite-field equation-of-motion (EOM) coupled-cluster (CC) implementation for electronically excited states in the coupled-cluster singles and doubles approximation (EOM-EE-CCSD) 32 giving access to excitation energies for atoms and molecules in strong magnetic fields. Obviously, besides excitation energies, for the generation of spectra also the dipole-oscillator strengths which correspond to given electronic transitions are required. In this work, we therefore present a scheme for the calculation of transition dipole moments in strong magnetic fields at the EOM-CCSD and CCSD linear response (LR) 33–35 levels of theory. The paper is organzied as follows: After a brief recapitulation of the theory with a focus on the calculation of properties in the two different schemes, we discuss the implementation and validation of the code subsequently. In the section on applications, we first present selected transitions of the sodium atom as a function of the magnetic field, together with corresponding (squared) transition dipole moments. For molecules, we discuss

3



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low-lying transitions in LiH in a magnetic field of B = 0 − 0.6 B0 both in the parallel as well as in the perpendicular orientation. Finally, we compare the EOM-CC results to their size-intensive CC-LR counterparts.

Theory In a uniform magnetic field, the electronic Hamiltonian for an N -electron molecule is N

N

X X ˆ =H ˆ0 + 1 H B · lîO + B · sî 2 i i N

+

2 1 X 2 O2 B ri − B · riO 8 i

(1)

ˆ 0 denoting the Hamiltonian for the field-free case. B is the vector of the magnetic with H field, sî the spin of electron i, riO its position vector with respect to the gauge origin O and lîO = −iriO × ∇i , the canonical angular momentum operator due to which the wave function may become complex. In EOM-EE-CC theory, 36,37 the final-state wave function for an electronically excited ˆ to the CC 38 ground-state state |Ψexc i is given by applying a linear excitation operator R wave function |ΨCC i

ˆ |ΨCC i = R ˆ eTˆ |Φ0 i , |Ψexc i = R

(2)

ˆ =R ˆ0 + R ˆ1 + R ˆ2 + . . . , R XX ˆn = 1 R rab... a ˆ† a ˆ† . . . a ˆj a î , (n!)2 ij... ab... ij... a b

(3)

where

(4)

and the indices i, j, . . . as well as a, b, . . . referring to occupied and virtual spin orbitals, respectively. |Φ0 i denotes the reference determinant, usually the Hartree-Fock determinant,

4


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and Tˆ = Tˆ1 + Tˆ2 + · · · =

X

tp τˆp

(5)

p

is the cluster operator with the cluster amplitudes tp and τˆp denoting strings of quasi-particle creation operators that generate excited determinants |Φp i when applied to the reference ˆ

determinant. Inserting Eq. (2) into the Schrödinger equation, pre-multiplying with e−T , and subtracting the energy ECC of the CC reference state leads to ¯ NR ˆ |Φ0 i , ˆ |Φ0 i = ∆Eexc R H

(6)

¯ N = e−Tˆ (H ˆ − ECC )eTˆ and ∆Eexc the energy with the non-Hermitian effective Hamiltonian H necessary to excite the system from the ground state |ΨCC i to the final state |Ψexc i. Using projection techniques, the eigenvalue problem can be formulated in an algebraic form ¯ N R = ∆Eexc R, H

(7)

¯ N matrix and the column vector R are given as where the elements of the H ¯ N |Φq i , ¯ N ]pq = hΦp |H [H

(8)

ˆ 0i , [R]p = hΦp |R|Φ

(9)

¯ N is non-Hermitian, the bra-state eigenvalue problem is not simply the respectively. Since H complex conjugate of Eq. (7) but instead given by ¯ N = ∆Eexc L, LH

(10)

where the elements of the row vector L are defined as ˆ pi [L]p = hΦ0 |L|Φ 5


(11)


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with the de-excitation operator Lˆ Lˆ = Lˆ0 + Lˆ1 + Lˆ2 + . . . , 1 X X ij... † † â ˆ ...a ˆb a â . l a Lˆn = (n!)2 ij... ab... ab... i j

(12) (13)

Accordingly, the vectors R and L are not orthogonal among themselves but biorthogonal among each other and can be chosen to obey ˆ (J) |Φ0 i = δIJ , L(I) · R(J) = hΦ0 |Lˆ(I) R

(14)

where I and J are state labels. While for excitation energies, the solution of the right-hand eigenvalue problem in Eq. (7) is sufficient, for properties, the solutions of both Eqs. (7) and (10) are required. Then, as shown in Ref. 36, excited-state dipole moments (I = J) and transition dipole moments (I 6= J) can be computed in a rather straightforward manner as expectation values ˆ ˆ ˆ (J) µI→J = hΦ0 |Lˆ(I) e−T µ ˆ eT R |Φ0 i

= Tr(ρI→J µ)

(15) (16)

by tracing the reduced one-particle density matrix ρI→J with elements ˆ † ˆ ˆ (J) [ρI→J ]rs = hΦ0 |Lˆ(I) e−T a ˆr a ˆs eT R |Φ0 i

(17)

with the dipole-moment integrals µ. The indices r and s refer to general molecular orbitals. Since EOM-CC is a non-Hermitian theory, the transition moments are not well defined. However, since the observable quantity is the dipole-oscillator strength fIJ which, in atomic

6


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Table 1: Expressions for the reduced one-electron density matrix in the spin-orbital basis for EOM-CCSD. The Einstein summation convention is used. ρij = ρab = ρia = ρai =

δIJ ·

tai

i h i h ef mi e f mi ef rm tj + 21 rmj + δij tmj − rje lei − lef −r0 lei tej + 12 lef ma e a 1 ea 1 mn ea a m mn r0 lb tm + 2 leb tmn + rm lb + leb rm tn + 2 rmn e mi i r0 la + rm lea i

h f ef a 1 mn e a ea a e + r0 lem (tea tea + l0 ria + lem (rmi − rm t − rie tam ) mi − ti tm ) − 2 lef mn ti + tmi tn i i h f ea ef a ea f e mn tfnia − tfi tan − 12 rna tef rm +lef mi + rmi tn + ri tmn + rmn ti

Note that for the CC bra reference state l0 = 1, otherwise l0 = 0. δij adds the HF contribution to ρij . ˆ and Lˆ correspond to the same state and δIJ = 0 for transition properties. δIJ = 1 if R

units, is given as 2 fIJ = (EJ − EI )|µI→J |2 , 3

(18)

only the unambiguous ’squared’ transition moments (STMs), i.e., the product of the ’left’ and ’right’ transition moment

|µI→J |2 = µI→J · µJ→I = Tr(ρI→J µ) · Tr(ρJ→I µ).

(19)

is required. 36 In the EOM-CC framework, transition properties are not size-intensive as the relaxation of the amplitudes in the reference wave function is not included. 36 To account for this amplitude-relaxation correction within the EOM-CC framework, the transition-dipole moment may be expressed in a connected manner 39,40 as h i ˆ (J) |Φ0 i µI→J = hΦ0 |Lˆ(I) µ ¯, R hh i i ¯ Tˆε (ωIJ ) , R ˆ (J) |Φ0 i , + hΦ0 |Lˆ(I) H,

(20)

ˆ ˆ ¯ ˆ ˆ Tˆ with the similarity-transformed dipole and Hamilton operators µ ¯ = e−T µ êT , H = e−T He

and the perturbed cluster operator Tˆε (ωIJ ) which depends on the frequency ωIJ , i.e., the 7



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difference between the total energies EI and EJ of the states |ΨI i and |ΨJ i. The expression in Eq. (20) may be also obtained as first derivative of the Lagrangian 39,40 h i ¯ R ˆ (J) |Φ0 i LIJ = hΦ0 | Lˆ(I) H, X (R) ˆ (J) − hΦ0 | Lˆ(I) PˆR ∆Eexc R |Φ0 i

(21)

R

with respect to the electric field ε. The first term in Eq. (21) corresponds to the connected EOM-CC energy expression while the second term is needed to ensure stationarity with ˆ thereby making use of the perturbation-independent respect to the amplitudes in Lˆ and R, projector defined by ˆ (R) |Φ0 i hΦ0 | Lˆ(R) . PˆR = R

(22)

In actual calculations, the determination of the perturbed CC amplitudes can be avoided by exploiting the (2n + 1)-rule 41 from derivative theory and the stationarity of the Lagrangian with respect to the CC amplitudes may be invoked by using the modified Lagrangian 39,40 ˜ IJ = LIJ + hΦ0 | Z(ω ˆ IJ ) H ¯ − ωIJ Tˆ |Φ0 i . L

(23)

The additional term adds the CC amplitude equations (including a frequency term arising due to the time derivative in the time-dependent Schrödinger equation) as constraints. Zˆ ij... is a de-excitation operator parametrized in the same manner as Lˆ with its amplitudes ζab...

representing the Lagrange multipliers: Zˆ = Zˆ1 + Zˆ2 + . . . 1 X X ij... † † Zˆn = ζ a â ˆ ...a ˆb a â . (n!)2 ij... ab... ab... i j

(24) (25)

˜ IJ with respect to the The transition-dipole moment is hence given by the derivative of L 8


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electric field ε:

µI→J =

˜ IJ dL dε

i h (J) (I) ˆ IJ )¯ ˆ ˆ |Φ0 i + hΦ0 | Z(ω µ |Φ0 i . µ ¯, R = hΦ0 | L

(26)

The first term can be evaluated in the same way as shown in (16) by using a modified EOMCC one-particle density matrix, i.e., by leaving out all terms including r0 and rem which ˆ represent disconnected contributions in Table 1. For the additional term, the Z-amplitudes have to be determined by solving the equations

0=

h ˜ IJ (J) i ∂L ¯ τˆp , R ˆ = hΦ0 | Lˆ(I) H, |Φ0 i ∂tp ˆ IJ ) H, ¯ τˆp |Φ0 i + hΦ0 | Z(ω ˆ IJ ) |Φp i . − ωIJ hΦ0 | Z(ω

(27)

It can be shown that, following this route, the obtained properties are equivalent to the ones obtained from CC-LR theory. 42

Implementation Taking advantage of the existing implementation of the left- and right-hand-side EOM-EECCSD eigenvalue problem as well as the contraction routine described in Ref. 32, one-electron properties have been implemented at the EOM-EE-CCSD and the CCSD-LR levels of theory. The necessary integrals over the gauge-including atomic orbitals, 43 the self-consistent field calculation using an unrestricted Hartree-Fock (UHF) reference wave function, 43 and the coupled-cluster amplitudes 44 are provided via an interface to the program package London. 45 In the case of EOM-EE-CCSD, the complex one-electron density is set up and then contracted with the required integrals. The implemented expressions for the reduced one-

9



electron density matrix are given in Table 1. For CC-LR-CCSD, the one-electron density ˆ is modified as discussed in the theory section. Additionally, the Z-equations are solved in an iterative manner in order to evaluate the second term in Eq. (20). The code has been validated in the following way: • For the field-free case, EOM-CC and CC-LR results 46 for transition moments have been compared to those calculated using the CFOUR program package. 47 • For cases where EOM-CCSD is equivalent to FCI, the results for transition moments in a magnetic field have been compared among the EOM-CC and CC-LR results and with the FCI results 48 provided by London. • For CC-LR, the perturbed CC amplitude equations were implemented to cross-check the result based on the implementations of Eqs. (20) and (26).

Applications Computational details In this section, we demonstrate the applicability of our implementation by investigating the evolution of STMs calculated at the EOM-CCSD level (see Eq. 19) as a function of the magnetic field for low-lying excited states of the sodium atom as well as for the LiH molecule. Furthermore, we investigate the importance of amplitude relaxation by comparing EOM-CCSD results to those from our CCSD-LR implementation. The magnetic field is varied between 0 and 0.5 B0 for sodium and up to 0.6 B0 for LiH. All calculations have been carried out using the uncontracted Cartesian aug-cc-pCVQZ basis set. 49 The use of large uncontracted basis sets is adequate when dealing with magnetic field strengths of less than 1 B0 . 44 In a magnetic field, the symmetry of atoms is lowered to C∞h . For molecules, the symmetry depends on the orientation with respect to the magnetic field. In the case of linear heteroatomic molecules, the symmetry is C∞ and Cs in the parallel and 10


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perpendicular orientations, respectively.1 Throughout the discussion, when referring to electronic states, the notation A/B is used with the state labels A and B denoting the symmetry of the system without and with the magnetic field, respectively. HF occupations are given using the field-free symmetry labels. The bond lengths for LiH have been determined for every calculated field strength in a perpendicular orientation at the CCSD level of theory with an uncontracted Cartesian aug-cc-pVQZ basis set. The same bond lengths have been used for the calculations in the parallel orientation. The complete list of energies, geometries, and dipole strengths can be found in the supplementary information (SI).

The sodium atom

-161.6

-161.8

Etot / Eh

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


12S/12Σg ([Ne] 3s) 12P/12Π−1,u ([Ne] 3p) 12P/12Σu ([Ne] 3p) 12P/12Π+1,u ([Ne] 3p) 22S/22Σg ([Ne] 4s) 12D/12∆−2,g ([Ne] 3d) 12D/12Π−1,g ([Ne] 3d) 12D/32Σg ([Ne] 3d)

-162.06 -162.08 -162.10 -162.12 0.0 0.05 0.1 0.15

-162.0

-162.2

0.0

0.1

0.2

0.3

0.4

0.5

B / B0

Figure 1: Total energy of the eight lowest doublet states of the sodium atom in a magnetic field. Inset: Avoided crossing between the 22 S/22 Σg and the 12 D/32 Σg states. Figures 1 and 2 show the total energies of the lowest doublet states of the sodium atom as a function of the magnetic field as well as a selection of STMs of the respective symmetryallowed transitions, i.e., those where the parity changes and ∆ML = 0, ±1, with ML denoting the total magnetic quantum number. From the total energies it is obvious that destabilizing diamagnetic and stabilizing paramagnetic contributions compete. As the selected states are 1

Note that in a perpendicular orientation the main symmetry axis is given by the magnetic field vector. As the latter is an axial vector, the mirror plane σh lies perpendicular to it and includes the molecular axis.

11



15

12S/12Σg → 12P/12Π±1,u 12S/12Σg → 12P/12Σu 12P/12Σu → 22S/22Σg 12P/12Σu → 12D/32Σg 12P/12Σu → 12D/12Π±1,g 12P/12Π±1,u → 22S/22Σg 12P/12Π±1,u → 12D/32Σg 12P/12Π±1,u → 12D/12Π±1,g 12P/12Π±1,u → 12D/12∆±2,g

14 13 12 11 10

µI→J · µJ→I / e2 a20

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9 8 7 6 5 4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

B / B0

Figure 2: EOM-CCSD STMs for selected dipole-allowed transitions between the eight lowest doublet states of the sodium atom in a magnetic field. doublets, and we have chosen the spin of the unpaired electron to be antiparallel with respect to the magnetic field, they all are stabilized by the spin-Zeeman term by -0.5 B0 . Further stabilization can occur from the orbital-Zeeman term for states with negative ML values (12 P/12 Π−1,u , 12 D/12 Π−1,g , and 12 D/12 ∆−2,g ). However, for increasingly strong magnetic fields, all selected states eventually rise in energy when the diamagnetic contribution becomes dominant. As the symmetry is lowered to C∞h by the presence of the magnetic field, certain states (i.e.,

2S+1

L and

2S+1

(L + 2) with the same ML ) may no longer cross as compared to

the field-free situation. For example, S states and D states with ML = 0 both become Σg states and may generate avoided crossings. Such an avoided crossing is visible, in Fig. 1 between the 22 S/22 Σg state shown in brown and the 12 D/32 Σg state in dark blue around 0.05 B0 (see inlet). By the interaction between the two states, one of them is stabilized while the other becomes destabilized and the character of the states interchanges. Turning to the respective STMs in Fig. 2, we note that they evolve in a rather complicated manner already for a simple atom. Nearly all STMs decrease (eventually) with the magnetic field strength. This behaviour could be due to the fact that as the field increases, the orbitals become compressed in the directions perpendicular to the magnetic field, diminishing the 12


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overlap. The most interesting behaviour is exhibited by transitions that involve the higherlying 22 Σg and 32 Σg states which undergo the avoided crossing mentioned before (shown with dotted lines). For all of these transitions, the STMs change drastically in curvature in the range where the avoided crossing occurs while the other states (solid lines) evolve in a rather smooth manner. For example, in the 3s → 3p transitions, the STM for 3s → 3p0 (solid red, 12 S/12 Σg → 12 P/12 Σu ) increases smoothly with the magnetic field, while the STM for the corresponding 3s → 3p±1 transition (solid light blue, 12 S/12 Σg → 12 P/12 Π±1,u ) smoothly decreases. However, the conceptually similar transitions 3p → 4s, i.e., those into the 22 Σg state are much more complicated: The STM for the transition from p0 (dotted brown, 12 P/12 Σu → 12 P/12 Σg ) also increases, though in a much steeper fashion, but then around 0.06 B0 turns to decrease again. The corresponding STM for the transition from p±1 (dotted yellow, 12 P/12 Π±1,u → 22 S/22 Σg ) decreases with a very similar, though negative slope towards zero and increases very slowly again for stronger magnetic fields. It is clear that in a strong magnetic field, the STM (and hence the intensity of a given transition) can by no means be inferred from its zero-field value. This is true even if one is interested only in a qualitative picture, i.e., strong versus weak transitions. Therefore, finitefield calculations of STMs will play an important role for the prediction and the assignment of spectra from magnetic white dwarfs.

The LiH molecule Figs. 3 and 4 show the evolution of low-lying singlet states of the LiH molecule as a function of a parallel and perpendicular magnetic field, respectively. In the parallel case (C∞ ) the selected states evolve in a rather simple manner. All states become destabilized by the diamagnetic term for a strong enough magnetic field. The 11 Σ+ /11 Σ and 21 Σ+ /21 Σ states (light blue and black) evolve essentially parallel as they are energetically too far apart to interact. The 11 Π/11 Π−1 state (red) is stabilized by the orbital-Zeeman term, while the 11 Π/11 Π+1 state (green) is destabilized by the same amount. As for the sodium atom, ML 13



is still a good quantum number in the magnetic field. Since there are no avoided crossings in the states under consideration the respective STMs evolve in a relatively simple manner (see Fig. 5), smoothly decreasing with the magnetic field. This situation is entirely different in a perpendicular magnetic field. As the symmetry is lowered to Cs , the Σ, Π, . . . symmetry is broken, such that these states will no longer cross2 but are instead allowed to mix, given that they transform in the same manner under reflection perpendicular to the magnetic field. This also means that the projection of the angular momentum is no longer quantized, i.e., ML is no longer a good quantum number. Therefore, by this interaction, the states can acquire angular momentum, which makes the situation more complex than in the case of atoms and linear molecules in parallel fields. Note that the perpendicular paramagnetic bonding mechanism 48 is a special case of such a paramagnetic interaction. A more detailed discussion on paramagnetic stabilization will be provided elsewhere. In the case of LiH, the 11 Σ+ /11 A0 ground state is not strongly affected, since it lies energetically much lower than the other states of A0 symmetry. It can however be noted, that it does not increase as steeply in energy as in the parallel case which suggests an interaction with the first excited 21 Σ+ /21 A0 state (light blue). The latter is stabilized for weak magnetic fields, showing that it must have admixed angular momentum. For stronger magnetic fields it is, however, destabilized and goes through a further avoided crossing with the 11 Π/31 A0 state (green). The 31 A0 state, originating from 11 Π with ML = +1 is additionally ’trapped’ by a higher-lying A0 state, originating from the 31 Σ state shown in grey. The other component of the Π state (ML = −1) is the only one of A00 symmetry for the states under consideration and increases smoothly with the magnetic field. The multitude of state interactions and avoided crossings becomes particularly obvious when turning to the STMs in Fig. 6. As expected from the discussion so far, the STM of the 11 Σ+ /11 A0 → 11 Π/11 A00 transition (red) is the only one that decreases smoothly with 2 Note that the rule that for a diatomic molecule all states of the same symmetry are not allowed to cross is no longer strictly valid, as there are now two degrees of freedom for a given direction of the magnetic field, i.e., the bond length and the field strength. However, the cases where states of the same symmetry do cross are so rare that they may safely be ignored in the present context

14


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the magnetic field. For all other transitions, the behaviour is quite complicated, showing pronounced peaks or dips in cases where avoided crossings occur. A similar observation has been made by Becken 50 in FCI calculations of the helium atom in a strong magnetic field. Note also that due to the reduced symmetry, the former 11 Σ+ → 11 Π transitions no longer give the same evolution of their respective STMs. Additionally, the former Π → Π transitions (brown) are symmetry-allowed in the perpendicular case such that there are six instead of three distinct transitions. -7.6

-7.7

Etot / Eh

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


-7.8

-7.9

11Σ+/11Σ (1σ 2 2σ 2) 21Σ+/21Σ (1σ 2 2σ 3σ) 11Π /11Π−1 (1σ 2 2σ 1π) 11Π /11Π+1 (1σ 2 2σ 1π)

-8.0

-8.1 0.0

0.1

0.2

0.3

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0.5

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Figure 3: Total energy of the four lowest singlet states of the LiH molecule in a magnetic field parallel to the molecular axis.

Comparison to CCSD-LR For the field-free case, it has been shown 51 that for a single small molecule the difference between EOM-CCSD and CCSD-LR transition properties is very small and the EOM-CC results are therefore sufficient. In order to investigate whether this also holds in strong magnetic fields, the STMs for excitations from the reference (11 Σ+ /11 A0 ) have been recalculated using our CCSD-LR implementation based on Eqs. (26) and (27). Fig. 7 shows the relative error of the EOM-CC results compared to the ones from CCSD-LR. The absolute value of the deviation depends on the particular state and field strength, but the errors are very 15



-7.6

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11 Σ+ /11 A0 21 Σ+ /21 A0 11 Π /11 A00 11 Π /31 A0 31 Σ+ /41 A0

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Figure 4: Total energy of the five lowest singlet states of the LiH molecule in a magnetic field perpendicular to the molecular axis.

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Figure 5: EOM-CCSD STMs for dipole-allowed transitions between the four lowest singlet states of the LiH molecule in a magnetic field parallel to the molecular axis.

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11 Σ+ /11 A0 → 21 Σ+ /21 A0 11 Σ+ /11 A0 → 11 Π /11 A00 11 Σ+ /11 A0 → 11 Π /31 A0 21 Σ+ /21 A0 → 11 Π/11 A00 21 Σ+ /21 A0 → 11 Π/31 A0 11 Π/11 A00 → 11 Π/31 A0

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0.2

0.3

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B⊥ / B0

Figure 6: EOM-CCSD STMs for dipole-allowed transitions between the four lowest singlet states of the LiH molecule in a magnetic field perpendicular to the molecular axis.

0.25

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∆EOM,rel (µI→J · µJ→I ) / %

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11 Σ+ /11 A0 → 21 Σ+ /21 A0 11 Σ+ /11 A0 → 11 Π /11 A00 11 Σ+ /11 A0 → 11 Π /31 A0

0.15

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Figure 7: Error of the EOM-CCSD dipole strength for transitions between the 11 Σ+ /11 A0 ground state and the three lowest excited singlet states of the LiH molecule in a magnetic field perpendicular to the molecular axis relative to CCSD-LR.

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small. For the considered field strengths the error is always smaller than 0.004 a.u. which is < 0.25% of the total STM. The same observations were also made for the 11 Σ+ → 11 Π transitions in CH+ (see SI). Therefore, using the EOM-CC expectation value approach seems adequate for the prediction of the intensity in spectra of small molecules in strong magnetic fields and a more expensive LR calculation is not needed.

Conclusion Transition-dipole moments have been implemented at the EOM-EE-CCSD and CCSD-LR levels of theory for the treatment of atoms and molecules in strong magnetic fields. The evolution of STMs in an increasingly strong magnetic field has been investigated for the sodium atom and for the LiH molecule in orientations parallel and perpendicular to the magnetic field. The EOM-EE-CCSD and CCSD-LR results are very similiar with a deviation of less than 0.25%, suggesting that, as in the field-free case, for a small systems the difference is neglible such that a treatment at the EOM-EE-CCSD level is sufficient. Since the dipole-oscillator strengths, or the intensity of the respective transition, are proportional to the STMs, the question of how the latter evolve in a magnetic field is directly related to whether or not a given transition will be visible in the spectra of magnetic white dwarfs. We observe a very complicated evolution of the STMs, even for a simple atom such as sodium. The complexity stems from avoided crossings which result from symmetry-lowering in the magnetic field. As a result, even the qualitative question whether a certain transition is weak or strong for a given range of magnetic-field strengths cannot be answered without resorting to finite-field calculations of STMs. Hence, for the assignment of spectra from white dwarf stars, finite-field EOM-EE-CCSD predictions of STMs will play an important role.

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Acknowledgement The authors thank J¨ urgen Gauss for fruitful discussions. This work has been supported by the Deutsche Forschungsgemeinschaft under the grant STO 1239/1-1. The authors are grateful to the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway, where parts of this work were carried out under the project ’Molecules in Extreme Environments’.

Supporting information Energies and STMs for sodium, LiH, and CH+ calculated at the EOM-EE-CCSD and CCSDLR level of theory including reference geometries. hSˆ2 i expectation values for the UHF reference for the sodium atom. EOM-EA-CCSD energies for the sodium atom. This information is available free of charge via the Internet at http://pubs.acs.org.

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12S/12Σg ([Ne] 3s) 12P/12Π−1,u ([Ne] 3p) 12P/12Σu ([Ne] 3p) 12P/12Π+1,u ([Ne] 3p) 22S/22Σg ([Ne] 4s) 12D/12∆−2,g ([Ne] 3d) 12D/12Π−1,g ([Ne] 3d) 12D/32Σg ([Ne] 3d)

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12S/12Σg → 12P/12Π±1,u 12S/12Σg → 12P/12Σu 12P/12Σu → 22S/22Σg 12P/12Σu → 12D/32Σg 12P/12Σu → 12D/12Π±1,g 12P/12Π±1,u → 22S/22Σg 12P/12Π±1,u → 12D/32Σg 12P/12Π±1,u → 12D/12Π±1,g 12P/12Π±1,u → 12D/12∆±2,g

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11 Σ+ /11 A0 → 21 Σ+ /21 A0 11 Σ+ /11 A0 → 11 Π /11 A00 11 Σ+ /11 A0 → 11 Π /31 A0 21 Σ+ /21 A0 → 11 Π/11 A00 21 Σ+ /21 A0 → 11 Π/31 A0 11 Π/11 A00 → 11 Π/31 A0

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