| Citation: |
Chaofan Li, Siting Hou, Changjian Xie. Three-Dimensional Diabatic Potential Energy Surfaces of Thiophenol with Neural Networks[J]. Chinese Journal of Chemical Physics , 2021, 34(6): 825-832. DOI: 10.1063/1674-0068/cjcp2110196
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The Born-Oppenheimer (BO) approximation [1] is the bedrock for understanding molecular spectroscopy and reaction dynamics. Due to large discrepancy in the masses of electron and nucleus, the motion of electrons and nuclei can be separately treated under the BO approximation. This thus leads to the concept of the BO potential energy surface (PES) [1], which depends parametrically on the nuclear coordinates. The BO approximation is commonly valid, yet breaks down as two electronic states are quite close or degenerate coincidently, in which nonadiabatic couplings have to be considered and a single PES is not adequate anymore in the description of nuclear dynamics. As a result, the multi-state coupled PESs are necessarily demanded in the accurate simulation of the nonadiabatic dynamics processes.
In particular, the nonadiabatic effects are introduced in the simulation by means of the multiple adiabatic PESs as well as the nonadiabatic couplings, or the diabatic coupled PESs alternatively [2]. Although two treatments in different (adiabatic and diabatic) representations are identical numerically [3, 4], the diabatic approach based on the diabatic PESs is preferable due to the fact that the diabatic potential energy matrix elements are all smooth in the whole configuration space of the nuclei and no divergence problems exist in the calculations [5]. Nevertheless, the rigorous diabatic PESs are built up from both the adiabatic energies and nonadiabatic couplings [6, 7], which is usually expensive and challenging. And the adiabatic and diabatic representations are connected by a unitary transformation [2].
To construct an analytic PES from the discrete ab initio energy points for a large nuclear configuration space, there is a wide variety of approaches, such as the spline interpolation [8, 9], many-body expansion (MBE) [10, 11], modified Shepard interpolation (MSI) [12, 13], reproducing kernel Hilbert space (RKHS) [14], interpolating moving least squares (IMLS) [15], permutation invariant polynomial (PIP) [16, 17], Gaussian process (GP) progression [18], and neural network (NN) methods [19-21]. Specifically, the NN method has merited remarkable attention due to its extremely large flexibility and high accuracy [22-24], which facilitates the development of sufficiently accurate PESs to perform accurate quantum or classical calculations. It is thus not surprising that the NN method has also been widely used to fit the multi-state coupled diabatic PESs of molecules [25-38], such as H3 [25, 36], LiHF [33, 35], ClH2 [32, 38], and NH3 [26, 31] systems.
Thiophenol has received increasing attention in the past several decades owing to its important role of the chromophores in the biological molecules [39-42]. The π∗←π transition in thiophenol is responsible for the strong ultraviolet (UV) absorption [41], and the photochemistry of thiophenol is dominated by the 1πσ∗-mediated photodissociation due to the repulsive nature of the πσ∗ state [41, 42]. Since there are 33 degrees of freedom (DOFs) in thiophenol, it is very challenging to construct the full dimensional PESs [43], especially using the high-level ab initio method. There have been two reduced diabatic PESs for thiophenol [44, 45], including the two-dimensional (2D) [44] and three-dimensional (3D) [45, 46] DOFs, which can be used in the photodissociation dynamics calculations. In previous 3D diabatic PESs of thiophenol [45, 46], the diabatization of PESs was achieved using the linear-vibronic coupling method [47, 48], in which the off-diagonal terms of the diabatic PESs were determined solely by the adiabatic electronic energies. And the PESs were finally fitted by the spline interpolation method. Based on the constructed 3D diabatic PESs, the calculated lifetimes of the S1 vibronic states of thiophenol were found to be in good agreement with experiment [46], which validated the reliability of the diabatic PESs. In this work, we constructed a new 3D diabatic PESs of the ππ∗ and πσ∗ states of thiophenol using the NN method based on the ab initio energy points and some more sampled points on previous 3D diabatic PESs [45]. We focus here on the strongly coupled conical intersection region between the 1ππ∗ and 1πσ∗ states, which locates nearby the Franck-Condon region and is responsible for the lifetimes of the S1 vibronic states. Thus, only the coupling between the 1ππ∗ and 1πσ∗ states was considered. The diabatization was achieved by the NN fitting with the off-diagonal term constrained by the correct symmetry of the system.
Since the diabatic potential energy matrix elements are smooth and continuous as a function of nuclear coordinates, in this work we used the NN functions to represent each element of diabatic potential energy matrix.
A multilayer forward-feed NN is built up of an input layer, one or more hidden layers, and an output layer. For a given layer (i+1), the form of the kth neuron can be written as [21, 49]
| yi+1k=fi+1(Ni∑j=1wi+1k,jyik+bi+1k) | (1) |
in which Ni is the number of neurons in the ith layer, wi+1k,j is the weights connecting the jth neuron in the ith layer and the kth neuron in the (i+1)th layer, and bi+1k is the biases of the kth neuron in the (i+1)th layer. fi+1 is the transfer function for the (i+1)th layer. It can be readily seen that the output of one layer is the input of the next layer. In our calculations, there are two layers in the hidden layer. The input vector includes three variables of the coordinates:
| G1=exp(−0.6R),G2=θ,G3=φ | (2) |
where R, θ, and φ (in Å, degree, degree) denote S-H stretch, C-S-H bend, and C-C-S-H torsion coordinates of thiophenol.
In this work, the diabatic PESs of the coupled excited states 1ππ∗ and 1πσ∗ were fitted by the NN method, and the single PES of the ground state 1ππ was fitted individually. The reliability of the new NN diabatic PESs was examined by comparing the calculated lifetime of the S1 vibronic state with that on previous diabatic PESs. The topography of the 1ππ∗ and 1πσ∗ PESs in Franck-Condon region is responsible for the lifetimes of the S1 vibronic states, since the S1/S2 conical intersection on the PESs of thiophenol locates at a short distance (R = 2.8 bohr). The rigorous construction of the diabatic PESs is based on the adiabatic energies and nonadiabatic couplings, which is very challenging for thiophenol due to quite many DOFs. Following earlier work [45, 46], here we build up the diabatic PESs merely based on the adiabatic energies with NN fitting method.
The 2×2 diabatic potential energy matrix for the 1ππ∗ and 1πσ∗ states can be written as
| V=(V11V12V21V22) | (3) |
Each matrix element is represented by a NN function in the fitting procedure. And the symmetry of each element has to be correctly considered. At the planar geometry (φ = 0), the 1ππ∗(V11) and 1πσ∗(V22) states are of A′ and A″ symmetries, respectively. But the symmetries of two states exchange at φ = 90∘. As a result, the coupling term V12 carries the A′×A″ = A″ symmetry. To correctly constrain the symmetry, the NN function for the coupling term V12 was multiplied by a factor sin2φ. Also the PESs are assumed to be symmetric with φ = 90∘ in the range of [0∘, 180∘] [46].
In the fitting procedure, the weights and biases of the NN functions are determined by minimizing the residual sum of squares:
| S=Ndat∑i=12∑n=1(Efitn,i−En,i)2 | (4) |
in which Ndat is the number of the energy points in the fitting. En,1 and En,2 are the adiabatic energies of S1 and S2 states, which can be easily obtained by diagonalizing the diabatic potential energy matrix in Eq.(3). The Levenberg-Marquardt algorithm [50] was used to iteratively optimize the weights and biases of the NN functions and the Jacobian matrix was calculated numerically by the central-difference algorithm with the displacements of 10−5.
In our calculations, 3952 adiabatic energy points were chosen from previous 3D diabatic PESs [46] to fit the ground state and the diabatic matrix elements of the 1ππ∗ and 1πσ∗ states of thiophenol. 16 points along R ranging from 1.0 Å to 2.5 Å with the interval of 0.1 Å, 13 points along θ in the range of [60∘, 130∘], and 19 points along φ in the range of [0∘, 90∘] with the interval of 5∘ were sampled.
In the NN fitting procedure, the transfer functions in the first and second hidden layers are the hyperbolic tangent function (f(x) = tanh(x)) and is a linear function in the output layer. 20 neurons in each of the two hidden layers were used in the NN fitting of the single S0 PES and the diabatic PESs of two coupled 1ππ∗ and 1πσ∗ states. FIG. 1 shows the root mean square error (RMSE) as a function of the iterative steps in the fitting for the S0, S1, and S2 states. It can be readily seen that the RMSEs quickly decrease below 10 meV for the two sets of the NN fitting. The convergence for a single PES fitting of S0 is achieved within 800 iterations and the RMSE is 2.1 meV. For the NN fitting of the diabatic PESs of 1ππ∗ and 1πσ∗ states, it converges quickly with 600 interactions and the RMSEs are 2.9 and 3.8 meV for the S1 and S2 states, respectively. FIG. 2 displays the fitting error distributions for the S0, S1, and S2 states. The distributions of the errors are quite balanced in the energy ranges (∼5 eV). The largest fitting errors for the S0, S1, and S2 states are found to be 13.9, 25.9, and 27.4 meV, respectively. Since the range of the energy points in the NN fitting is not large, the weight factors for the energies in the NN fittings are not included.
To examine the accuracy of the NN PES of the S0 state, we calculated the vibrational energy levels of the excited states for the three vibrational modes (C-C-S-H torsion, C-S-H bend, and S-H stretch). The Lanczos method was used to obtain the 3D vibrational energy levels and the grids in the calculations were the same as those described in Ref.[45]. As shown in Table Ⅰ, the calculated energy levels on the new NN PES are 70, 970, and 2478 cm−1, respectively, which are in accord with those on the previous spline PES (75, 972, and 2499 cm−1). Good agreement of the vibrational energy levels with those on previous PES validates the accuracy of new NN PES.
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FIG. 3 shows the one-dimensional (1D) adiabatic potential energy curves along R for four different φ values (0∘, 30∘, 60∘, and 90∘) of the S0, S1, and S2 states with the C-S-H angle fixed at the equilibrium (θ = 96.2∘) of S0 state. As expected, both the NN PESs of the single S0 and the coupled S1 and S2 states are fitted very well. The S1/S2 conical intersection at φ = 0∘ is located at R = 2.795 bohr on the diabatic PESs, which is in excellence with the location (R = 2.786 bohr) of the conical intersection on previous diabatic PESs [46]. FIGs. 4 and 5 display the 3D plot of the adiabatic PESs of the S1 and S2 states as a function of R and φ with θ = 96.2∘ and of θ and φ with R = 2.80 bohr, respectively, fitted by the NN method, and the spline PESs from Ref.[46] are also shown for comparison. The locations and shapes of the conical intersections are clearly shown in FIGs. 4 and 5. It is clear that the NN PESs reproduce the topographies of the spline ones especially for the conical intersection regions quite well, which validates the high accuracy of the NN approach.
In FIG. 6, the 1D NN diabatic potential energy curves along R for four different φ values (0∘, 30∘, 60∘, and 90∘) of the 1ππ∗ and 1πσ∗ states (θ = 96.2∘) are compared to the spline ones. It can be seen that the 1ππ∗ state is very close for the NN and spline PESs. For the 1πσ∗ state, two sets of the PESs are similar with some differences for longer R>3.0 bohr. The differences should be caused by two aspects. One is different diabatization methods utilized for two sets of PESs. Furthermore, the coupling between the 1ππ and 1πσ∗ states at longer R was considered in the spline PESs [46], while it was not included in our NN PESs. FIG. 7 displays the 3D plot of the diabatic PESs of the 1ππ∗ and 1πσ∗ states (θ = 96.2∘) along R and φ fitted by the NN method. It shows that the NN and spline diabatic PESs are quite similar as discussed above.
To further validate the reliability of the new NN diabatic PESs of the 1ππ∗ and 1πσ∗ states, the energy and lifetime of the zero-point energy (ZPE) state 00 of the S1 state of thiophenol were calculated on the new NN PESs. The calculated energy and lifetime are 40935.5 cm−1 and 25.9 fs, respectively. Both of the two are found to reproduce the results (40957.2 cm−1 and 28.2 fs) [46] on the previous spline PESs quite well, which suggests the conical intersection regions of the 1ππ∗ and 1πσ∗ states are well represented by the NN approach. On the other hand, the diabatization of previous spline PESs was achieved by the linear-vibronic coupling method [46], while the NN diabatic PESs in this work were done by the fitting method with the correct symmetry constraint on the off-diagonal term. Good agreement between two sets of the PESs indicates the reasonable treatment of the nonadiabatic coupling of the conical intersection region of the 1ππ∗ and 1πσ∗ states by two diabatization approaches.
To summarize, in this work we report the NN method to calculate the single S0 PES and the diabatic PESs of two coupled 1ππ∗ and 1πσ∗ states of thiophenol. The potential energy points were chosen from previous spline PESs [46] in the fitting, and no ab initio calculations were done in this work. As expected, the NN method behaves quite well on the fitting of the single PES of S0 and the RMSE is 2.1 meV. The calculated low-lying energy levels of the S0 state on the new NN PES are found to close to those on previous PES, which suggests the high accuracy of the NN method. Specifically, for the the NN fitting of the diabatic PESs of 1ππ∗ and 1πσ∗ states, the adiabatic energies were merely included. The nonadiabatic coupling was not included but with the correct symmetry constraint on the off-diagonal term. The RMSEs are 2.9 and 3.8 meV for the S1 and S2 states, respectively. The energy and lifetime of the ZPE state of the S1 state were calculated on the new NN diabatic PESs, which are in good agreement with previous spline diabatic PESs, validating the reasonability of the diabatic PESs constructed by the NN diabatization method.
This work was supported by the National Natural Science Foundation of China (No.22073073). Changjian Xie thanks the Startup Foundation of Northwest University. The Double First-class University Construction Project of Northwest University is acknowledged.
†Part of Special Issue "John Z.H. Zhang Festschrift for celebrating his 60th birthday".
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