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Excited States of Molecules in Strong Uniform and Non-Uniform Magnetic fields Sangita Sen,1, A) Kai K

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Excited States of Molecules in Strong Uniform and Non-Uniform Magnetic fields Sangita Sen,1, A) Kai K Excited states of molecules in strong uniform and non-uniform magnetic fields Sangita Sen,1, a) Kai K. Lange,1 and Erik I. Tellgren1, b) Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway (Dated: 1 February 2019) This paper reports an implementation of Hartree-Fock linear response with complex orbitals for computing electronic spectra of molecules in a strong external magnetic fields. The implementation is completely general, allowing for spin-restricted, spin-unrestricted, and general two-component reference states. The method is applied to small molecules placed in strong uniform and non-uniform magnetic fields of astrochemical importance at the Random Phase Approximation level of theory. For uniform fields, where comparison is possible, the spectra are found to be qualitatively similar to those recently obtained with equation of motion coupled cluster theory. We also study the behaviour of spin-forbidden excitations with progressive loss of spin symmetry induced by non-uniform magnetic fields. Finally, the equivalence of length and velocity gauges for oscillator strengths when using complex orbitals is investigated and found to hold numerically. I. INTRODUCTION employed to enforce gauge-origin invariance and accel- erate basis set convergence27–30. With ordinary Gaus- External magnetic fields can dramatically affect the sians it becomes necessary to use very large basis sets to 31–35 electronic structure of atoms and molecules when approach gauge-origin invariance . An implementa- the field interaction strengths are comparable to the tion of integral evaluation for the LAOs which are plane- 36–39 Coulomb interaction1,2. This turns out to be of the order wave/Gaussian hybrid functions is this necessary . of 1 a.u ≈ 235 kT. In nature, such strengths are known Our implementation builds on our previous work on non- 40 to exist on magnetized stellar objects such as magne- uniform magnetic fields and General Hartree–Fock the- 41 36,42 tized white dwarf stars but they are two-three orders of ory within the London program . Since only the magnitude beyond what can presently be produced in one-electron part of the Hamiltonian is modified in such terrestrial experiments3–5. The observed electronic spec- a finite-field approach, no additional effort is required for tra from magnetized white dwarf stars are strongly dis- extension to post-Hartree–Fock theories or for linear re- torted by the magnetic fields making them impossible to sponse, in this case. It therefore opens up the possibility interpret without computational support. He, C and O of studying non-perturbative phenomena. have been detected so far, in addition to H6–8. Recently, Linear response provides computationally cheap ac- 9 H2 has been detected in non-magnetized white-dwarfs . cess to a large number of excited states. This is The possibility of small hydrocarbons cannot be ruled beneficial for the interpretation of complicated spectra out either10. The first computational efforts primarily where a large number of states are involved. While by Ivanov and Schmelcher11–14 were targeted at ground the role of differential electron correlation between the and excited states of small atoms at the Hartree-Fock ground and excited state is certainly important, com- level. Later work focussed on few electron systems such putational results in the literature (without magnetic as H2, He, He2, Li and Be at the full configuration in- fields) have clearly demonstrated that linear response teraction (FCI) level15–22. Most recently, the coupled- spectra are adequate in most cases if the ground-state is cluster theory (CCSD) has been used to compute ground well described as in coupled cluster linear response (CC- states of atoms and molecules in strong magnetic fields23 LRT)43–46, time dependent density functional theory followed by the equation-of-motion coupled cluster treat- (TD-DFT)47,48 or multi-configurational time-dependent ment (EOM-CCSD) for excited states24. Hartree-Fock (MCTDHF)49. It has been recently shown In this paper we present the first implementation of the that excitation energies from the Random Phase Approx- linear response of the Hartree-Fock method with complex imation (RPA) correspond to an approximated EOM- orbitals for computation of electronic spectra in an ex- CCD50. For example, in this paper we have demon- arXiv:1901.11086v1 [physics.chem-ph] 30 Jan 2019 ternal magnetic field. Earlier work in the non-relativistic strated that the evolution of the spectra of the carbon domain has focussed on spin frustrated systems25,26. The atom with changing magnetic fields is qualitatively very ground state is optimized in the presence of an exter- similar to the EOM-CCSD results by Hampe and Stop- nal magnetic field and the excited states are obtained kowicz24. via linear response. London atomic orbitals (LAOs) are In addition to the possibilities for supporting spectral detection of atoms and molecules in stars, the study of excited states in strong magnetic fields is also an unex- plored field as of today. Non-perturbative transition from a)Electronic mail: [email protected] closed-shell para- to diamagnetism51 and a new bonding b)Electronic mail: [email protected] mechanism15,52–54 in very strong magnetic fields have re- 2 cently been computationally uncovered for ground states. then write The usually more sensitive electronic structure of excited 1 1 states gives rise to the possibility of discovering inter- Atot(r) = B × rg − rh × (C × rh); (4) esting field-induced phenomena at field strengths lower 2 3 than that for ground states. Moreover, the response of 1 Btot(r) = B + C × rh: (5) a molecule to a magnetic field is found to increase with 2 increase in the area of cross-section perpendicular to the field51. This entails computations on excited states of Furthermore, the constant vector encoding the anti- larger molecular systems which become accessible to us symmetric part of b equals the curl of the magnetic field, with the linear response technique. Excited states also r × Btot = C. provide a wider range of possible electronic structures than ground states. Our implementation is entirely general and is able to B. Linear Response Formulation handle non-uniform fields which break spin-symmetry, necessitating a two-component representation of orbitals Due to the loss of time reversal symmetry, Hartree- even with a non-relativistic Hamiltonian41. We can Fock (HF) computations for atoms and molecules in thus study how the spin-forbidden excitations behave finite magnetic fields require complex-valued orbitals. with a progressive loss in spin-symmetry. In particu- Thus, the exposition below gives a general formulation for lar, we study the lowest singlet-triplet transition for var- complex-valued orbitals without recourse to assumptions ious molecules in this paper. The behaviour of oscillator of purely real or purely imaginary quantities. A gen- strengths is also investigated. eral non-orthonormal basis55–58 (e.g., the atomic orbital basis) is allowed in the derivation and implementation, although the reported applications have been carried out in the orthonormal molecular orbital (MO) basis. II. THEORY AND IMPLEMENTATION y The creation operator a^α creates an electron in the spinorbital α, while the annihilation operator a^α annihi- A. The Hamiltonian lates such an electron. Letting Sαβ denote the inverse y of the overlap matrix Sβγ = hvacja^βa^γ jvaci, it is now The non-relativistic Schrödinger–Pauli Hamiltonian, possible to define which is used in this work, is given by (in atomic units) αy αβ y α βα a^ = S a^β; a^ =a ^βS : (6) 1 X 2 X X 1 X H^ = π^l − v(rl)+ + Btot(rl)·S^l (1) 2 rkl Note the implicit summation over β in the above expres- l l k<l l sions. Multiplication by the overlap matrix yields where π^ = −ir +A (r ) is the mechanical momentum l l tot l a^y = S a^αy; a^ =a ^αS : (7) operator. γ γα γ αγ We choose a linearly varying non-uniform magnetic Borrowing terminology from differential geometry, in- field, in general, which can be written in the form dices occur in both covariant (subscript) and contravari- ant (superscript) positions59,60. We rely on the summa- T 1 Btot(r) = B + rh b − rh tr(b); (2) tion convention that indices that occur in both positions 3 are summed over, unless otherwise indicated. Unitary or orbital invariance is ensured when all contractions are of where B is a uniform (position independent) component, this form. In general, contraction with the overlap ma- b is a 3 × 3 matrix defining the field gradients, and trix or its inverse lowers and raises indices, respectively, r = r − h is the position relative to some reference h in the manner seen above. Clearly, the distinction be- point h. This form may be viewed as arising from a Tay- tween covariant and contravariant indices disappears in lor expansion around r = h truncated at linear order. an orthonormal basis, where both the overlap matrix and The corresponding vector potential can be written as its inverse equals the identity matrix. We also note that a generic second-quantized 1-particle 1 1 T Atot(r) = B × rg − rh × (rh b); (3) operator A^ has the form 2 3 A^ = Aαβa^y a^ : (8) where rg = r − g, g being the gauge origin. It can be α β verified that Btot = r × Atot and that the magnetic By contrast, a generic 1-particle reduced density operator field is divergence free, r · Btot = 0. In what follows, we quantify the non-uniformity of the field through the is of the form anti-symmetric part C = b of the matrix b and α αβγ βγ ^ y αβ take the symmetric part, b = bT , to vanish.
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