Accurate Computation of Quantum Excited States with Neural Networks

Abstract

We present a variational Monte Carlo algorithm for estimating the lowest excited states of a quantum system which is a natural generalization of the estimation of ground states. The method has no free parameters and requires no explicit orthogonalization of the different states, instead transforming the problem of finding excited states of a given system into that of finding the ground state of an expanded system. Expected values of arbitrary observables can be calculated, including off-diagonal expectations between different states such as the transition dipole moment. Although the method is entirely general, it works particularly well in conjunction with recent work on using neural networks as variational Ansätze for many-electron systems, and we show that by combining this method with the FermiNet and Psiformer Ansätze we can accurately recover vertical excitation energies and oscillator strengths on a range of molecules. Our method is the first deep learning approach to achieve accurate vertical excitation energies, including challenging double excitations, on benzene-scale molecules. Beyond the chemistry examples here, we expect this technique will be of great interest for applications to atomic, nuclear and condensed matter physics.

Figures (11)

• Figure 1: Excited state energies for first row atoms from lithium to neon. Results from natural excited state VMC applied to the FermiNet (10 states, blue, 5 states, red) are shown on top of experimental resultssansonetti2005handbook. Spectral lines which match computed states are labeled with electron configurations and atomic term symbols (except for the highest levels of F and Ne, where term symbols are omitted for clarity). For all but the largest systems and highest excited states, there is excellent agreement with experiment. The discrepancy between 5 and 10 excited states is minimal except for the highest excited states of F and Ne, where computing more states increases the accuracy of a given state. Complete numerical results are given in Table \ref{['tab:atomic_spectra']}.
• Figure 2: Vertical excitation energies and oscillator strengths for small molecules. Singlet states are in blue and triplet states are in gray. NES-VMC results are indicated by markers while theoretical best estimates from Chrayteh et al.chrayteh2020mountaineering or directly from QUEST veril2021questdb are given by the lines. When no data from QUEST is available, no TBE is given. Experimental results from Chrayteh et al.chrayteh2020mountaineering and references thererin are given by the dashed lines in green. Where available, energies and oscillator strengths from Entwistle et al.entwistle2023electronic are provided by the black triangles for comparison, with (pointing left) and without (pointing right) variance matching. In almost all cases, our results on both energies and oscillator strengths agree closely with theoretical best estimates. Complete numerical results are given in Table \ref{['tab:oscillator_strengths']}.
• Figure 3: Excited states of the carbon dimer (C$_2$).(a) Potential energy curves of the low-lying excited states of C$_2$, smoothed by cubic interpolation. The dotted lines are calculated by SHCI holmes2017excited, shifted by 0.115 Ha to account for basis set effects. (b) The vertical and adiabatic energies of excited states of C$_2$. The green line indicates experimental adiabatic energiesmartin1992c2, the orange line indicates adiabatic energies from SHCI holmes2017excited and the red line indicates the vertical energy of the $B^1\Delta_g$ state from QUESTloos2019reference. Bright transitions are labelled with their oscillator strength and, when available, their names. (c) The symmetries of the different states can be identified by evaluating each single state Ansatz at location $\mathbf{r}$ and $-\mathbf{r}$ for parity symmetry (u/g, blue) or by flipping $\mathbf{r}$ across the x-axis for reflection symmetry (+/--, orange). (d) Visualization of the 8 lowest natural orbitals of C$_2$. (e) The occupancy of the different natural orbitals for the different excited states of C$_2$, identified from the density matrix of each state. The $a^3\Pi_u$ through $A^1\Pi_u$ states are single excitations while the $b^3\Sigma^-_g$ and $B^1\Delta_g$ states are double excitations. Complete numerical results are given in Tables \ref{['tab:carbon_dimer']}, \ref{['tab:carbon_dimer_pec']} and \ref{['tab:carbon_dimer_adiabatic']}.
• Figure 4: Excited states and conical intersection of ethylene (C$_2$H$_4$).(a) Potential energy curve of the first two singlet states of ethylene under torsion around the C-C bond. Our results (blue) are compared against TD-DFTmalis2020trajectory (purple), MR-CIbarbatti2004photochemistry (green) and a penalty method used with the PauliNet, without the variance matching correctionentwistle2023electronic (red). (b) Potential energy curve of the first two singlet states of ethylene under pyramidalization of the C-H bonds. The best estimate of the location of the conical intersection of the V and N states for each method is given by the vertical line. Our method is in close agreement with MR-CI up to a constant shift, and agrees with the location of the conical intersection better than the PauliNet penalty method. Note that the $\phi=0$ geometry in Fig. \ref{['fig:ethene']}b differs slightly from the $\tau=90$ geometry in Fig. \ref{['fig:ethene']}a, as in Barbatti et al.barbatti2004photochemistry. All results are normalized so that the ground state energy at the equilibrium geometry is 0. Complete numerical results are given in Table \ref{['tab:ethene']}.
• Figure 5: Excited states of larger double excitation systems. NES-VMC with FermiNet and Psiformer for singlet (blue) and triplet (gray) systems are compared against results from the QUEST databaseveril2021questdb (singlet single excitations in blue, triplets in grey, singlet double excitations in red), including "unsafe" results (dashed lines). For systems where the QUEST results are "unsafe", more accurate results from DMC (orange) shepard2022double or with a CASPT3 correction (purple) kossoski2024reference are given. NES-VMC closely matches the more accurate double excitation calculations on the largest and most challenging systems. Complete numerical results are given in Table \ref{['tab:double_excitations']}.