Learning Full Configuration Interaction Electron
Correlations with Deep Learning

Hector H. Corzo∗
hhcorzo@gmail.com

Arijit Sehanobish†‡arijit.sehanobish1@gmail.com

Onur Kara§okara83@gmail.com

Abstract
In this report, we present a deep learning framework termed the Electron Correlation Potential Neural Network (eCPNN) that can learn succinct and compact
potential functions. These functions can effectively describe the complex instantaneous spatial correlations among electrons in many–electron atoms. The eCPNN
was trained in an unsupervised manner with limited information from Full Configuration Interaction (FCI) one–electron density functions within predefined limits ofaccuracy. Using the effective correlation potential functions generated by eCPNN,
we can predict the total energies of each of the studied atomic systems with aremarkable accuracy when compared to FCI energies.

1

Introduction

Applications of machine learning (ML) are ubiquitous in many branches of the natural sciences.
In particular, deep neural networks (NN) are revolutionizing the approach in which problems inmolecular sciences are explored. ML techniques offer an attractive alternative for reducing thecomputational time, hardware demands, and the complexity needed for many–electron atom andmolecular simulations. A multitude of propositions using ML for the treatment of such quantum
systems exist. NN frameworks have been constructed for atomistic simulations[1, 2], end-to-end
learning of the forward and inverse Schrödinger equation[3, 4], molecular dynamics simulations[5,6, 7, 4], and the direct representation of highly correlated wave functions [8, 9, 10]. Such studies
inspired the design of an unsupervised learning model capable of defining effective functions thatcan describe many–electron systems. By using limited information obtained from the electronic
probability density, a NN is constructed that is able to learn an effective Hamiltonian which describes
the electronic correlation effects relevant for many–electron atoms. This NN approach for describing
many–electron systems has been developed on the intrinsic conceptual formalism of the configurationinteraction (CI) method for solving non-relativistic many–electron correlated systems within the
Born–Oppenheimer approximation. High fidelity CI data was used to directly train the NN in an
unsupervised manner.
This work focuses on three main points, (a) the introduction of a NN model for many-electron atoms
defined in terms of FCI concepts and high fidelity data, (b) the approximation of observables such as
total energy for atomic system and (c) the agreement of our learned potential with the virial theorem,
which although originally formulated for classical systems, has been shown to hold for quantum
mechanical systems as well [11, 12, 13].
∗

Center for Chemical Computation and Theory, University of California, Merced, CA.
Work done as a postdoc at Yale University. Currently at Covera Health, NYC‡
Equal Contribution§
Hindsight Technology Solutions†

Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

2

Correlated Electronic Densities for Atomic Systems

In atomic and molecular theory, numerically exact solutions to the electronic time-independent, nonrelativistic Schrödinger equation for many–electron systems is given by the configuration interaction
matrix-eigenvalue equation [14, 15, 16],HCµ = EµF CI Cµ ,(1)

abc...
where H is the representation of the Hamiltonian Ĥ in terms of Slater determinants Dijk...
, Cµ
F CI
are known as the variational coefficients, and Eµ
is the total energy of the system within an
infinitely complete basis set. Beyond one electron systems, the operator Ĥ is not separable, thus only
approximate solutions for equation 1 exist. If a complete set of orthonormal one-electron functions ofspace-spin coordinates χ = (~r, ξ) is defined, every normalizable antisymmetric wave-function can beabc...
expressed as a linear combination of Dijk...
. Then, the FCI wave-function written as an expansion ofabc...
variational coefficients (cijk... ), in its cluster form, reads
XX
XX
X X
ab ab
abc abc
Dia cai +
Dij
cij +
Dijk
cijk + · · · ,(2)
ΨF CI = D0 c0 +iai<j a<bi<j<k a<b<cwhere the indices i, j, k denote occupied (Occ) spin-orbitals or Dirac bi-spinors, whereas the a, b, c
indices represent the correspond unoccupied or virtual (Virt) one-electron functions. In the ΨF CI ,the CI space is expanded according to electron substitutions or excitations levels from Occ to Virt
orbitals defining different configurations. In this respect, non–relativistic selected configurationinteraction (SCI) wave–functions, ΨSCI [15, 17, 18] provide compact wave-functions to definereliable and stable electronic densities for many–electron systems over a wide range of values of theradial coordinate (~r). By definition, ΨSCI yields equivalent energies as ΨF CI in a more compact
representation where only the most important configurations within a predefined limits of accuracy
are computed [17, 15]. ΨSCI are defined by a finite expansion of configuration state functions (CSFs)
expressed as linear combination of Slater determinants,
X p
ΨSCI =
φK aKp ,(3)
K,pwhere φpK aKp represents the CSFs with K configurations and p degenerate elements. After unitary
SCItransformation and diagonalization, the electronic probability density of Ψ
terms of natural radial orbitals, χil (r), and occupation numbers nil ,Xρ(r) =f (i, l, ms)nil χ2il (r),may be expressed in(4)iwhere f (i, l, ms) is the LS symmetry factors of the CSFs.
For this study the ground state atomic densities for Lithium 2 S (Li), Beryllium 1 S (Be), and Neon 1 S(Ne) were computed with an average of 484 CI term expansion for the interacting and non–interacting
spaces, and a Slater type orbital (STO) basis with harmonics in a 0 ≤ l ≤ 5 range. All densities(fig. 1) were defined in terms of the density matrix using the Slater–Condon rules for monoelectronic
operators and only the LS eigenfunctions that correspond to the leading CSFs within a threshold
contribution of 0.1 µHa for each of the ground states were considered.

3

Electron Correlation Potential Neural Network (eCPNN)

The Hamiltonian operator Ĥ, as defined in the previous section, is often conceptualized as the sumof the kinetic (T̂) and potential energy (V̂) operators. An alternative, equally valid conceptualizationof Ĥ is in terms of a fluctuation operator, (Ĥ − Ĥ0 ), which contain all the information regarding the
instantaneous spatial correlations among all electrons, Thus, in atomic units (a.u.),~2 ∂ 2~2 2+ V̂(~r) ≡ −∇ + V̂(~r).
(5)
22m ∂~r2m ~r
Generally, the fluctuation or interactive term (or V̂) is unknown and difficult to compute [17], whereas
the non-interacting term Ĥ0 (or T̂) may be inferred using the second derivatives of the correlated FCI
probability amplitude with respect to its spatial coordinates.
Ĥ = T̂ + V̂ = Ĥ0 + (Ĥ − Ĥ0 ) = −2

Figure 1: Atomic probability densities for A) Lithium 2 S; B) Beryllium 1 S; C) Neon 1 S.
An alternative way to characterize quantum systems is presented by reformulating quantum phenomena as a solution of inverse problems (i.e, approximating an effective function containing all theimportant physical constrains that generated the observed outcomes). In this context, the probability√
densities (ρ) of multi-electronic systems can be used to define a proxy function (|φ| = ρ) to
describe the quantum system. Although this proxy does not contain the complete knowledge of all theelectronic interactions and correlations, its information is enough to define a NN that can construct
an effective Hamiltonian to accurately describe the quantum system under study. We propose to train
our physics informed deep learning model [19, 20, 21] via the following equation
 2

~2 ∇~r2 |φ|LEC (θ) = Dr −,(6)
+ Uθ (~r)
2m |φ|
2
where Dr is the total derivative operator (with respect to ~r) acting on multi-variate function
2
~2 ∇~
r |φ|
r), || · ||2 is the Frobenius norm, and Uθ the learned parametric function that− 2m
|φ| + Uθ (~approximates the electronic correlation for each many–electron system. Because Uθ is given by adifferential equation, the eCPNN is capable of approximating the effective potential V (~r) for atomic
systems up to an arbitrary constant. An initial condition is introduced to ensure the uniqueness ofthe solution via the Picard–Lindelöf theorem. Equation 6, termed the effective correlation (EC) loss
function, not only obeys the theoretical referents of the FCI matrix-eigenvalue equation (Eq. 1), buteffectively demands energy conservation for the many–electron quantum systems. Thus, for the
proposed model, the complete loss function reads
L(θ) = LEC (θ) + (Uθ (~r) − y)2 ,(7)

where ~r is some point in the domain of the function and y is the expected ground truth value for thetrue correlation function at that point. It is important to point out that using |φ| instead of ΨF CI for
solving the electronic correlation problem may lead to the incorrect fluctuation potential at finitely
many points where ΨF CI changes signs. However, this last does not cause any difficulties in training
our network.
In our experiments, the eCPNN model is a 3 layer feedforward network with hidden sizes of 64, 128
and 128 with tanh non–linearity within the hidden layers and a residual connection between thesecond and third layer. The inputs to the model are the ~r spatial coordinates. For the network training,
5000 of these coordinates were randomly selected from the domain of definition of each atomic
system. The model was trained for 500 epochs with Adam optimizer [22] and a learning rate of 0.001.
For stable training, we used a L2 regularization on the weights of the network with the weight tuning
parameter of 0.0001.
Our code is available at https://github.com/arijitthegame/Quantum-Hamiltonians.

4

Experiments

The eCPNN model was validated on the Li, Be and Ne atoms. For all the atoms, spatial coordinates
are defined on the interval [0.1, 10]. As explained in equation (7), an appropriate initial conditionis added for each atom to ensure that an unique potential function is learned. The learned effective
potential functions for Li, Be and Ne atoms present a maximum whose height and width correlatewith the number of electrons in each atom (fig. 2). The larger the number of electrons the taller and3

Figure 2: Learned correlation potentials (top), Learned energies (middle), virial theorem deviation(bottom).
more narrow the maximum in the potential curve. Using our learned potential function and the precomputed density, we estimate the total energy of the system. Table 1 shows the FCI energies (EF CI )for each of the atoms studied and the energies approximated by the eCPNN model. For the atoms
SystemLithium Atom
Beryllium Atom
Neon Atom

EF CI

EeCP N N

RMSE: True vs. Learned Energies−7.477487−14.66101−128.888004−7.5431−14.6142−128.8584

0.07 ± 0.04
0.01 ± 0.02
0.01 ± 0.009

Table 1: FCI and eCPNN Learned Energies in a.u.: The 2nd column shows the average energy
calculated by eCPNN over 5 runs.

studied, the largest error in the eCPNN predicted energy correspond to Li. Unlike the Be and Ne atoms
with closed–shell reference configurations for the ground state, 1s2 2s2 and 1s2 2s2 2p6 respectively,
the reference configuration for Li is 1s2 2s1 which correspond to an open–shell configuration with aunpaired electron in a s shell. It is well known that computing the energy of open–shell systems ismore challenging due to their electronic structure characterized by having different number of α andβ electrons. Thus, different considerations for the exchange and correlation electronic effects areneeded. However, for the closed–shell atoms, the energies computed by eCPNN are remarkably close
to the true energies (E F CI ). In the learned V (~r), the attractive regions at small ~r present an upturn
close to the origin. The sharp maximum in V (~r) seems to be an attribute of the potential where its
position coincides with the minimum in 4πr2 ρ(~r) separating the peaks in the density from the atomic
shells. These results suggest that the peak in V (~r) separates spatially the parallel spin electrons in the
different shells. For more details, see Appendix A.1.
In the context of density functional theory (DFT), the virial theorem is commonly used as a constraint
to calculate atomic energies within the Kohn-Sham DFT formalism and it is frequently utilized as an4indicator of the quality of the approximation in computational chemistry tasks [11, 23, 24, 25, 26, 27].
As shown in fig. 2, for all the studied atoms, the effective potentials learned by the eCPNN alsosatisfy the virial theorem, 2hT i + hV i = 0.

5

Conclusion

We present a deep learning framework called eCPNN which was trained using high fidelity FCI data
in an unsupervised manner. The eCPNN can approximate compact potential functions capable ofdescribing the complex electric correlation effects important for describing atomic systems. This
presents the possibility of defining succinct functions capable of predicting highly accurate energies
for other many–electron atomic systems. Furthermore, we show that our learned potential functions
satisfy important physical constraints like the virial theorem. Our work opens up interesting research
directions in various problems in catalysis, rovibrational spectroscopy, photoelectron spectroscopy,
and excited-state chemistry and enzymatic processes. An important drawback of our study is thedependency of the defined potentials on the calculated high fidelity densities. Studies are underway
to create more robust systems when a measurable amount of noise is present.

6

Broader Impact

We envision this work to be beneficial to a broader community since we hope it will encourage
researchers to use deep learning in trying solve various complicated differential equations. We are
however limited by the curse of dimensionality as it will be significantly difficult to be able to run
these experiments on a CPU.
Quantum mechanics has been one of the most successful models for describing the physical world.
However, quantum mechanical systems are generally hard to solve and exact solutions only exist for
simple systems. As such, by leveraging the power of neural networks we have aimed to improve
the practical use of quantum mechanics and thus potentially contribute to the understanding of ourworld. One caveat is that our method provides only an approximation for the description of quantum
phenomena, and thus the possibility of incorrect predictions cannot be precluded.

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6

A

Appendix

A.1

Electron Density

Mathematically, the function ρ(r) exhibits non-analytical behaviour in the form of a discontinuityof its gradient exactly at the origin r = 0, (i.e. the positions of the nuclei) which results poles in
the potential energy function at these positions (V = −∞)[28]. The electronic wave–function mustsatisfy the cusp condition (which result in spikes of ρ(r) here) in the neighbourhood of each of
the nuclei electronic wave–function, where ρ(r) ∼ e−2Zr . Note: How sharp the spikes observed
around the cusp depends on the charge of the nucleus Z such that an infinitesimal deviation fromthe position of the nucleus has to be accompanied by such a decreasing of the density given bylimr→0 d lndrρ(r) = −2Z. Figure 3 illustrates a contour map of the electron density distribution ina plane containing the nucleus for the n = 1 level of the H atom. The distance between adjacentcontours is 1 atomic unit (a.u.). The numbers on the left-hand side on each contour correspond to theelectron density in a.u. The numbers on the right-hand side correspond to the proportion of the total
electronic charge which lies within a sphere of that radius. The figure therefore reflects the fact that
99% of the single electronic charge of the H atom lies within a sphere of radius 4 a.u.

Figure 3: Cusp behavior of ρ near r=0

7

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