Hartree Energy Functional

• Hartree energy functional is a mathematical model that captures classical electron-electron Coulomb repulsion in density functional theory.
• It underpins variational analysis by ensuring convexity and stability of electronic systems, with applications in ground-state determination and time-dependent dynamics.
• Matrix representations and numerical optimization frameworks using LCAO basis sets enable precise benchmarking of exchange–correlation models in quantum many-body computations.

The Hartree energy functional is a cornerstone of both rigorous quantum many-body analysis and practical electronic structure computation. It encodes the classical electrostatic (Coulomb) repulsion between continuous charge densities, arising as a universal term in density functional theory (DFT), as the electron–electron interaction contribution in Hartree(-Fock) theory, and as a convolution-type energy in nonlinear dispersive PDEs. Its mathematical structure underlies both qualitative features such as convexity and stability, and precise tools for testing and benchmarking approximate exchange–correlation models.

1. Definition and Formulation

The archetypal Hartree energy functional is defined for a charge density $n(r)$ on $\mathbb{R}^3$ by

$$E_H[n] = \frac{1}{2} \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \frac{n(r)\,n(r')}{|r - r'|} \, dr \, dr' .$$

This form, universally adopted in DFT (Joubert, 2011), ensures symmetrization of the bilinear interaction and avoids double-counting via the factor of $\frac12$. In nonlinear PDE setting, for $u \in H^1(\mathbb{R}^3)$, the Hartree energy reads

$$\mathcal{E}(u) = \frac{1}{2}\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx + \frac{1}{4} \iint_{\mathbb{R}^3 \times \mathbb{R}^3} |u(x)|^2 w(x-y) |u(y)|^2 dx \, dy,$$

with $w$ encoding the interaction kernel, allowing for generalizations such as Coulombic or more singular forms (Pistillo, 18 Dec 2025).

2. Functional Derivatives: Potential and Kernel

The first functional derivative yields the Hartree potential,

$$V_H(r) = \frac{\delta E_H[n]}{\delta n(r)} = \int_{\mathbb{R}^3} \frac{n(r')}{|r - r'|} dr',$$

representing the electrostatic potential generated by $n(r)$ (Joubert, 2011). When the Hartree term is combined with exchange–correlation terms, the total functional is $F[n] = E_H[n] + E_{xc}[n]$.

The second functional derivative defines the kernel

$$\frac{\delta^2 F[n]}{\delta n(r)\,\delta n(r')} = \frac{1}{|r - r'|} + f_{xc}(r,r'),$$

with $$f_{xc}(r, r') = \delta^2 E_{xc}/\delta n(r)\delta n(r')$$ the exchange–correlation kernel. The Hartree contribution is strictly repulsive and positive-definite, guaranteeing convexity (Joubert, 2011). The stability of electronic systems thus depends on the properties of $f_{xc}(r, r')$, as negative curvature may indicate unphysical behavior in approximate functionals.

3. Ground State Existence and Variational Structure

The variational problem for the Hartree energy, particularly in the critical case $d=3$ with an admissible interaction $w$ satisfying $w = w_1 + w_2$, $w_1 \in L^\infty$, $w_2 \in L^{3/2,\infty}$, seeks minimizers $u$ under a fixed $L^2$-mass constraint: $$I(\lambda) = \inf_{u \in S_\lambda} \mathcal{E}(u), \quad S_\lambda = \{u \in H^1(\mathbb{R}^3) : \|u\|_{L^2}^2 = \lambda \}.$$ Existence of ground states is guaranteed for $\lambda_* < \lambda < \lambda^*$, where $\lambda_*$ and $\lambda^*$ derive from non-positivity and functional estimates involving $w_2$ and a key Lorentz-convolution constant $K$ (Pistillo, 18 Dec 2025). Attainment, regularity, and positivity of ground states is established using concentration–compactness, strict subadditivity (binding), and elliptic regularity arguments. Ground states solve a constrained Euler–Lagrange equation: $-\Delta u_* + (w * |u_*|^2) u_* = \omega u_*.$ Here, $\omega < 0$ acts as a chemical potential, enforcing the mass constraint.

4. Matrix Representation and Numerical Optimization in LCAO Basis

In Hartree-Fock and closed-shell settings, the Hartree energy functional appears as part of a fourth-order polynomial in the coefficients of a linear combination of atomic orbitals (LCAO): $$E[\{c\}] = \sum_{i,\mu,\nu} c_{\mu i} c_{\nu i} h_{\mu\nu} + \frac12 \sum_{ij,\mu\nu\lambda\sigma} c_{\mu i}c_{\nu j}c_{\lambda i}c_{\sigma j}[(\mu\nu|\lambda\sigma) - (\mu\sigma|\lambda\nu)],$$ subject to orthonormalization constraints on the molecular orbitals (Kristyan, 2018). Optimization of $E[\{c\}]$ under these constraints is recast as a Lagrangian minimization, leading to a generalized eigenvalue problem for the Fock matrix, and can be addressed via Newton–Raphson iterative methods utilizing gradient and Hessian of the Lagrangian. This approach yields quadratic convergence near the optimum, contrasting with self-consistent field (SCF) eigensolvers commonly used in electronic structure calculations.

5. Identities, Curvature, and Consequences for Density Functional Theory

A central result concerning the Hartree energy’s role in DFT is the sum-rule for the second derivative of $F[n] = E_H[n] + E_{xc}[n]$, as demonstrated in Joubert (Joubert, 2011): $$\int d^3r\,[n_N(r) - n_{N-1}(r)] \frac{\delta^2 F[n_N]}{\delta n(r) \delta n(r')} = \text{constant}, \quad \forall r',$$ where $n_N$ and $n_{N-1}$ are ground-state densities for $N$ and $N-1$ electrons (same Hamiltonian). This identity reflects a sum-rule: the curvature of $F[n]$ weighted by the density difference (localized in the region of the “removed” electron) is spatially constant, linking to the derivative discontinuity in the exchange–correlation potential at integer electron number. This provides a stringent benchmark for the accuracy of approximate exchange–correlation kernels and connects to the prediction of ionization potentials and chemical hardness.

The positivity of the Hartree kernel ensures strict convexity of $E_H[n]$ and stability against unphysical instabilities, provided the exchange–correlation kernel preserves positive semi-definiteness (Joubert, 2011).

6. Dynamical Equations and Orbital Stability

The time-dependent Hartree (TDH) equation,

$$i \partial_t \psi(t,x) = -\Delta_x \psi(t,x) + (w * |\psi(t,\cdot)|^2)(x) \psi(t,x),$$

evolves initial data in $H^1(\mathbb{R}^3)$ and conserves both mass and Hartree energy. For admissible potentials $w$ and smallness conditions on the initial mass and $\|w_2\|_{L^{3/2,\infty}}$, global-in-time well-posedness is established. Furthermore, in mass ranges where ground states exist, the set of minimizers is orbitally stable: solutions starting sufficiently close (in $H^1$) to the ground-state manifold remain close for all time. This result leverages the variational structure, conservation laws, and compactness arguments (Pistillo, 18 Dec 2025).

7. Analytical and Technical Estimates

Rigorous analysis of the Hartree energy, both for existence and dynamical properties, employs functional-analytic techniques adapted to the convolution structure and allowed singularities in $w$. Essential estimates include Lorentz-space Hölder and Young inequalities to control convolutions of $w$ with $|u|^2$, as well as regularity bootstrapping and symmetrization for radial potentials. In matrix-based implementations, the Hessian of the Lagrangian and block-structure in terms of basis set integrals enable efficient Newton–Raphson optimization, and stabilization of iterative minimization.

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The Hartree energy functional thus provides not only the basic model for electron–electron repulsion and self-consistent electrostatics but also a rigorous framework for variational analysis of electronic ground states, benchmarks for functional derivatives, and a critical testbed for both analytic functional theory and computational electronic structure methods (Joubert, 2011, Kristyan, 2018, Pistillo, 18 Dec 2025).