In this study, we use the simplest model, Fermi’s Golden rule (FGR) based approach, in the calculation of relaxation rate. Because presumably, the relaxation rate of our system depends on weak vibronic coupling. Nevertheless, for strong vibronic coupling systems, one can construct the diabatic effective Hamiltonian following the approach we shown and the MCTDH wavepacket method can be applied in simulation of dynamics.
The construction of effective Hamiltonian is not an on-the-fly method We consider dy-namics in the way that proper for the Hamiltonian (weak vibronic coupling). Qualit-atively, diabatic state is electronic state that does not change its physical character as one moves along a reaction coordinate[61, 62]. In contrast to the adiabatic state, which changes constantly so as to remain eigenstates of the electronic Hamiltonian. One classic
doi:10.6342/NTU201803678 example of the interplay between the adiabatic and diabatic picture is the dissociation of
sodium-chloride (NaCl). As shown in Fig. 2.1, the ground adiabatic state is thought of as arising from the avoided crossing between an ionic and a covalent state. As the bond gets shorter, this ground state transforms from Na-Cl to Na+-Cl−. The adiabatic states thus change character, while the diabatic states keep their ionic and covalent character respectively.
The picture of diabatic electronic states takes advantages in many chemical phenomena.
Adiabatic Diabatic
Internuclear Distance (R)
Energy
Req Rǂ
𝑁𝑎+ + 𝐶𝑙−
𝑁𝑎 + 𝐶𝑙
Figure 2.1: Potential energy curves for the diatomic NaCl.
For example, diabatic states can be used in the construction of potential energy surfaces because they are smooth functions of the nuclear coordinates[63, 64, 65]. In spectro-scopy, diabatic states help to assign vibronic transitions and rationalize the rates of inter-state transitions[66, 67, 68]. Since typically, the diabatic inter-states have a small derivative coupling, which simplify the description of electronic transitions, diabatic basis is good for describing the electronically excited dynamics[69, 70, 71, 72, 73]. Additionally, dia-batic states connect to clearly defined product channels in scattering theory[74]. Finally, diabatic states play a qualitative role in our understanding of molecular bonding[75] (as briefly discussed by the NaCl example above), electron transfer[76], and proton tunneling[77,
doi:10.6342/NTU201803678 78, 79].
2.3.1 Definition of Diabatic Basis
A set of electronic states is defined as strictly diabatic states, ˜Φm, only if the derivat-ive coupling between any two states vanish at every possible nuclear configuration (any geometry). That is,
for any i, j, and ~R. This definition is in line with the qualitative concept that the electronic wavefunctions of diabatic states do not change when the nuclear geometry changes.
For a given set of M adiabatic states, Φm, it is essential to contruct a set of M strictly diabatic states which span the same Hilbert space as the given adiabatic states. The chosen electronic states are the states involved in a particular reaction one aims to study. In other words, one would like to have a set of orthonormal states
Φ˜m =
M
X
n=1
Amn( ~R)Φn (2.26)
that satisfy Eq. (2.25). The matrix A is the adiabatic-to-diabatic transformation matrix, or diabatization matrix.
If states of this form could always be obtained, they would clearly provide the rigorous definition of diabatic states. Simply specifying the set of interesting adiabatic states, the corresponding diabatic states could be automatically determined[80, 81]. Unfortunately, it is not generally applicable to create an strictly diabatic basis (SDB) from a given adiabatic basis, as in a famous proof given by Mead and Truhlar[82].
2.3.2 Nonexistence of Strictly Diabatic Basis for General Molecules
The derivative coupling between two adiabatic states is defined as
F~mn( ~R) ≡ −1
doi:10.6342/NTU201803678 Once the adiabatic states are chosen, ~F is fixed and cannot be changed. One can write
Eq. (2.25) in terms of ~F and the diabatization matrix, A:
which can be considered as the condition that determines the A once the couplings in adiabatic basis are known[80, 81]. We now take the derivatives of Eq. (2.29) with respect to R. After some manipulation, we obtain
∇ × ~F = ~F × ~F . (2.30)
This relationship is called the curl condition, and specifies a restriction that must be satis-fied by ~F if one hopes to construct an SDB out of the given adiabatic basis. Unfortunately, this condition is not satisfied for the Born-Oppenheimer eigenstates of real molecules, ex-cept under rare circumstances[82, 83].
Given the nonexistance of strictly diabatic states for general cases, one must weaken the search criteria in some ways to obtain diabatic states for a particular application. This is the so-called quati-diabatization which we will briefly introduce.
2.3.3 Quasi-diabatization Methods Overview
Quasidiabatic states, as opposed to strictly diabatic ones, are defined as states in which the derivative coupling does not vanish completely, but remains small. There is, of course, an ambiguity in what one means by small and as a consequence many techniques have been developed to construct such quasidiabatic states[84]. These quasidiabatic states are defined aim to have pratical use in some certain real systems. Common to most of these approaches is the idea to avoid the calculation of the derivative coupling elements altogether. Rather, one looks for suitable molecular properties and requires that they de-pend only weakly on the nuclear coordinates. For example, Werner and Meyer have used the eigen functions of the dipole operator as quasidiabatic basis states[85], Buenker et al.
employed transition dipole elements for their construction[86]. Hendekovi´c proposes that
doi:10.6342/NTU201803678 the diabatic basis maximizes the sum of squared occupation probabilities of natural spin
orbitals[87]. More recently, the block diagonalization scheme for diabatization has been proposed[88]. Finally, Atchity and Ruedenberg applied the principle of configurational uniformity to a quantum chemical determination of diabatic states[89, 90].
Broadly, these approaches can be divided into two categories, one tries to define the best set of diabatic states from a given set of adiabatic state (deductive strategies), and the other one attempts to construct the diabatic states directly (constructive strategies).
One of the well known diabatization methods is the Mulliken-Hush approach for electron transfer[91, 92]. In this method, the approximate diabatic states are defined using purely the information of spectroscopically observable (transition energy, transition dipole, and change in dipole). This approach can be generalized to deal with multiple states in a way that only involves adiabatic quantities if one defines the diabatic states as the eigenstates of the dipole moment operator[93, 94]. Similarly, problems with multiple charge center or excitation energy transfer can be dealt with by defining the diabatic wavefunctions with maximally localization in real space. That is, in such methods, the eigenstates of certain physical observable (such as the dipole of localization) will not change much as molecu-lar geometries rearrange. The physical picture of the problem are invoked and applied to define the diabatic basis for electron- and energy-transfer systems.
One class of constructive strategies is the density constraint approach. The quasi-diabatic states are obtained by optimizing the wavefunction subject to a constraint on the density.
This concept is the basis of the frozen density functional method[95, 96] as well as the constrained density function theory (CDFT) approach[97].
Bearing these developments in mind, in this study, we adopt the idea of the enforcing configuration uniformity proposed by Atchity and Ruedenberg. These slowly varying diabatic states are predominantly constructed from fixed set of electronic configurations.
In the next section, we will introduce the concept and theory of this method thoroughly.
doi:10.6342/NTU201803678