Diabatic-At-Construction Method for Diabatic and Adiabatic Ground and Excited States Based on Multistate Density Functional Theory

Abstract

We describe a diabatic-at-construction (DAC) strategy for defining diabatic states to determine the adiabatic ground and excited electronic states and their potential energy surfaces using the multistate density functional theory (MSDFT). The DAC approach differs in two fundamental ways from the adiabatic-to-diabatic (ATD) procedures that transform a set of preselected adiabatic electronic states to a new representation. (1) The DAC states are defined in the first computation step to form an active space, whose configuration interaction produces the adiabatic ground and excited states in the second step of MSDFT. Thus, they do not result from a similarity transformation of the adiabatic states as in the ATD procedure; they are the basis for producing the adiabatic states. The appropriateness and completeness of the DAC active space can be validated by comparison with experimental observables of the ground and excited states. (2) The DAC diabatic states are defined using the valence bond characters of the asymptotic dissociation limits of the adiabatic states of interest, and they are strictly maintained at all molecular geometries. Consequently, DAC diabatic states have specific and well-defined physical and chemical meanings that can be used for understanding the nature of the adiabatic states and their energetic components. Here we present results for the four lowest singlet states of LiH and compare them to a well-tested ATD diabatization method, namely the 3-fold way; the comparison reveals both similarities and differences between the ATD diabatic states and the orthogonalized DAC diabatic states. Furthermore, MSDFT can provide a quantitative description of the ground and excited states for LiH with multiple strongly and weakly avoided curve crossings spanning over 10 Å of interatomic separation.

Figure 1

Adiabatic potential energy curves for the four lowest ¹Σ⁺ states of LiH as functions of interatomic distance from MSDFT (dashed curves) using the PBE0 density functional and from MS-CASPT2//SA(4)-CASSCF(2,9) calculations (solid curves). The aug-cc-pVTZ basis set with the diffuse s-function replaced by that of the Dunning-Hay basis (simply denoted by aug-cc-pVTZ throughout) is used in all MSDFT computations, whereas the standard aug-cc-pVTZ basis is used in wave function methods.

Figure 2

Adiabatic potential energy curves for the four lowest ³Σ⁺ triplet states of LiH (a, c, d, and e in order of increasing energy) as functions of interatomic distance from MSDFT (solid curves) using the PBE0 density functional and from MS-CASPT2/SA(4)-CASSCF (dashed curves). The aug-cc-pVTZ basis set is used in all computations (see caption of Figure 1).

Figure 3

(a) Valence-bond diabatic (solid curves) and adiabatic (dashed curves) potential energy curves of LiH as a function of interatomic separation optimized by multistate density functional theory (MSDFT). The PBE0 density functional along with the aug-cc-pVTZ basis set (see Figure 1) is used in all calculations. (b) Diabatic potential energy curves obtained by the three-fold way diabatization approach. Adapted from Figure 5 of ref with permission of the American Institute of Physics.

Figure 4

Orthogonalized diabatic (dashed curves) potential energy curves of LiH as a function of interatomic separation, transformed from the nonorthogonal VB-diabatic states in Figure 3(a) by (a) direct Gram-Schmidt (GS) orthogonalization, (b) GS-projection of mixed of 2s and 2p covalent states, (c) 2s-2p mixing scaled by the overlap integral, and (d) as in (c) along with a small amount of 25% 2s state added to the ionic state. The adiabatic potential energy curves are shown as solid curves.

Figure 5

Squared diabatic couplings between selected pairs of orthogonal diabatic states of LiH shown in Figure 4(d) as a function of interatomic separation. All results are determined by MSDFT with the aug-cc-pVTZ basis set (see Figure 1).

Figure 6

Computed derivative coupling values, ⟨Ψ_I(r;R)|∇_R|Ψ_J(r;R)⟩ for various states indicated as a function of interatomic separation. All results are determined by MSDFT with the aug-cc-pVTZ basis set (see Figure 1).