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We calculate the A.C. optical response to circularly polarized light of a Weyl semimetal (WSM) with varying amounts of tilt of the Dirac cones. Both type-I and II (overtilted) WSM are considered in a continuum model with broken time reversal (TR) symmetry. The Weyl nodes appear in pairs of equal energies but of opposite momentum and chirality. For type-I the response of a particular node to right (RHP) and left (LHP) hand polarized light are distinct only in a limited range of photon energy Ω, 2 1+C 2 /v < Ω µ < 2 1−C 2 /v with µ the chemical potential and C2 the tilt associated with the positive chirality node assuming the two nodes are oppositely tilted. For the over tilted case (type-II) the same lower bound applies but there is no upper bound. If the tilt is reversed the RHP and LHP response are also reversed. We present corresponding results for the Hall angle.

We use a Kubo formalism to calculate both A.C. conductivity and D.C. transport properties of a dirty nodal loop semimetal. The optical conductivity as a function of photon energy Ω, exhibits an extended flat background σ BG as in graphene provided the scattering rate Γ is small as compared to the radius of the nodal ring b (in energy units). Modifications to the constant background arise for Ω ≤ Γ and the minimum D.C. conductivity σ DC which is approached as Ω 2 /Γ 2 as Ω → 0, is found to be proportional to √with vF the Fermi velocity. For b = 0 we recover the known three-dimensional point node Dirac resultof Γ (universal) and the ratioπ 2 where all reference to material parameters has dropped out. As b is reduced and becomes of the order Γ, the flat background is lost as the optical response evolves towards that of a three-dimensional point node Dirac semimetal which is linear in Ω for the clean limit. For finite Γ there are modifications from linearity in the photon region Ω ≤ Γ. When the chemical potential µ (temperature T ) is nonzero the D.C. conductivity increases as µ 2 /Γ 2 (T 2 /Γ 2 ) for µ Γ T Γ ≤ 1. Such laws apply as well for thermal conductivity and thermopower with coefficients of the quadratic law only slightly modified from their value in the three-dimensional point node Dirac case. However in the µ = T = 0 limit both have the same proportionality factor of √ Γ 2 + b 2 as does σ DC . Consequently the Lorentz number is largely unmodified. For larger values of µ > Γ away from the nodal region the conductivity shows a Drude like contribution about Ω ≅ 0 which is followed by a dip in the Pauli blocked region Ω ≤ 2µ after which it increases to merge with the flat background (two-dimensional graphene like) for µ < b and to the quasilinear (three-dimensional point node Dirac) law for µ > b.

We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of non-adiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wave functions. ADT matrices eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrödinger equation to its diabatic form. Two real molecular systems NO and CH (Bz) are selectively chosen for the demonstration of workability of those equations. We examine the NACTs among the lowest five electronic states of the NO radical [X̃A (1B), ÃE″ (1A and 1B) and B̃E' (1A and 2B)], in which all types of non-adiabatic interactions, that is, Jahn-Teller (JT) interactions, Pseudo Jahn-Teller (PJT) interactions, and accidental conical intersections (CIs) are present. On the other hand, lowest five electronic states of Bz [X̃E (1B and 1B), B̃E (1A and 1B), and C̃A (1B)] depict similar kind of complex feature of non-adiabatic effects. For NO radical, the two components of degenerate in-plane asymmetric stretching mode are taken as a plane of nuclear configuration space (CS), whereas in case of Bz, two pairs are chosen: One is the pair of components of degenerate in-plane asymmetric stretching mode, and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asymmetric stretching modes. We calculate ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chemistry package. Subsequently, the ADT is performed using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, symmetric, and continuous diabatic PESs for both the molecular systems.

The major bottleneck of first principle based beyond Born−Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO 2 radical, H 3 + , F + H 2 , NO 3 radical, C 6 H 6 + radical cation, and 1,3,5-C 6 H 3 F 3 + radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.0 (GPLv3) is available at https://github.com/AdhikariLAB/ADT-Program and also as the Supporting Information of this article.

First principles based beyond Born–Oppenheimer theory has been implemented on the F + H2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin–orbit (SO) coupling effect on the collision process of the F + H2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (12A′, 22A′, and 12A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn–Teller (JT) type conical intersections between the two A′ states translate along C2v and linear geometries in F + H2. In addition, A′ and A″ states undergo Renner–Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H2, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H2 system.

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrödinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C6H3F3+ (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃2E′′, Ã2A2′′, B̃2E′, and C̃2A2′), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C–C asymmetric stretching, C–C symmetric stretching, and C–C–C scissoring motion (Q9x, Q9y, Q10x, Q10y, Q12x, and Q12y) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs.