Optimizing Nonlinear Variational Parameters

The screened-exponent helium calculation is not a historical curiosity; it is the direct conceptual seed of modern quantum chemistry software. Every time a program like Gaussian, ORCA, or Psi4 runs a Hartree–Fock or density functional calculation, it is solving a vastly more elaborate version of exactly this problem: choosing a basis of functions with adjustable exponents, and letting the total electronic energy determine both the linear expansion coefficients (via the machinery that becomes Linear Variation and Secular Equations) and, in fully optimized basis sets, the nonlinear exponents themselves. In drug design, effective nuclear charges and orbital contraction directly govern electronegativity trends used to predict hydrogen-bond donor and acceptor strength in candidate molecules — the same screening logic explaining why helium's electrons see an effective charge of $1.6875$ rather than $2$ explains why oxygen's lone pairs are more diffuse than fluorine's, shaping how a drug molecule's functional groups interact with a protein binding pocket. In materials science, the same idea underlies pseudopotential theory used in plane-wave density functional calculations of semiconductors and battery materials: chemically inert core electrons are replaced by an effective, parameter-optimized potential that reproduces the correct screening of valence electrons without explicitly treating every core orbital. In spectroscopy, Slater-type and Gaussian-type orbital exponents optimized this way feed directly into predicted absorption and emission energies used to assign unknown spectral lines in astrophysics and atmospheric chemistry — ionization energies and orbital sizes computed from optimized exponents underlie models of stellar atmospheres and planetary photochemistry. Even in undergraduate laboratory instruction, the helium exponent-optimization calculation remains one of the few calculations a student can perform entirely by hand while still touching a genuinely unsolved three-body problem, making it a perennial bridge between exactly solvable textbook physics and the approximate machinery that underlies essentially all of practical computational chemistry.

Ask the tutor: How would the helium calculation change if the two electrons were allowed to have independently optimized exponents instead of a single shared value?

Start from the variational theorem: for any normalized, well-behaved trial function $\psi_{\text{trial}}$ satisfying the correct boundary conditions, the expectation value of the true Hamiltonian obeys

where $E_0$ is the true ground-state energy. This inequality holds for any choice of trial function — including one built from a family of functions labeled by a continuous parameter $\zeta$. If you write $\psi(\zeta)$ for such a family, then $E(\zeta) = \langle\psi(\zeta)|\hat H|\psi(\zeta)\rangle$ is itself a function of $\zeta$, and the theorem says every value of this function lies above $E_0$. The best member of the family is whichever $\zeta$ makes $E(\zeta)$ smallest — this is now an ordinary one-variable calculus minimization, not a functional variation over all possible wavefunctions. The physical intuition is that $\zeta$ typically controls how tightly or loosely the electron density clings to the nucleus. A hydrogen-like 1s orbital $\psi = N e^{-\zeta r}$ has a decay constant $\zeta$: large $\zeta$ means a compact, tightly bound-looking cloud; small $\zeta$ means a diffuse cloud. The kinetic energy contribution to $E(\zeta)$ grows as $\zeta^2$ (steep functions have more curvature, hence more kinetic energy by the uncertainty principle), while the potential energy contribution — attraction to the nucleus — grows more favorable (more negative) linearly in $\zeta$ for a Coulomb potential. These two competing trends produce a single minimum: too small a $\zeta$ wastes potential energy by keeping the electron too far from the nucleus, too large a $\zeta$ wastes kinetic energy by over-localizing it. The compromise point is $\zeta_{\text{opt}}$. For helium, the same balance operates but with an added twist: the two electrons repel each other. Using a product trial function of two hydrogen-like orbitals sharing one common exponent $\zeta$, that exponent no longer represents the bare nuclear charge $Z=2$ — it represents an effective nuclear charge, reduced because each electron feels the true nucleus screened by the average presence of the other. The electron-electron repulsion integral, when included in $E(\zeta)$, shifts the minimum from $\zeta=2$ down to a smaller, screened value. This screening concept — later systematized into Slater's rules and into modern effective-core-potential methods — first appears here as a rigorous number from a two-line derivative, not as a rule of thumb.

The general recipe for nonlinear parameter optimization rests on three equations.

where: $\psi(\zeta)$ is the trial function built with adjustable parameter $\zeta$; $\hat H$ is the exact Hamiltonian of the system; the denominator handles normalization explicitly in case $\psi(\zeta)$ is not pre-normalized for every $\zeta$. Reading: this is simply the variational energy functional, now viewed as an ordinary function of the single real number $\zeta$ rather than as a functional over an infinite-dimensional space of functions.

where: the derivative is taken holding all other structural choices (functional form, basis) fixed; $\zeta_{\text{opt}}$ is the stationary point, and for well-behaved unimodal $E(\zeta)$ curves (as in these examples) it is the unique minimum. Reading: optimizing a nonlinear variational parameter is literally freshman calculus applied to a quantum mechanical expectation value — the hard work is entirely in constructing $E(\zeta)$ correctly.

where: $E_0$ is the true (generally unknown) ground-state energy of the exact Hamiltonian; $E(\zeta_{\text{opt}})$ is the best energy this particular trial family can produce. Reading: the theorem never breaks; it simply tells you that the minimum over $\zeta$ is the tightest bound this trial form is capable of giving. A different functional form for $\psi$ could still do better, even with more parameters or none at all — the optimization only searches within the chosen family. For a Coulombic one-electron problem the energy functional built from a hydrogen-like 1s trial function with exponent $\zeta$ always separates cleanly into a kinetic piece scaling as $\zeta^2$ and a potential piece scaling linearly in $\zeta$:

where: the $\zeta^2/2$ term is the kinetic energy expectation value (in hartree, atomic units), the $-Z\zeta$ term is the electron-nucleus attraction expectation value, and $Z$ is the nuclear charge felt directly by the electron in the Hamiltonian (not necessarily equal to $\zeta$ itself). Reading: this quadratic-in-$\zeta$, linear-in-$Z\zeta$ structure is the single most important pattern to memorize; it recurs for hydrogen exactly and for helium's screened problem in modified form.

Goal. Starting from the trial function $\psi = N e^{-\zeta r}$ for the hydrogen atom's single electron, derive $E(\zeta)$ in closed form, minimize it over $\zeta$, and confirm that the variational method recovers the exact hydrogen ground state exactly — the essential sanity check before trusting the method on helium.

Step 1 — Set up the Hamiltonian and trial function. In atomic units ($\hbar = m_e = e = 4\pi\varepsilon_0 = 1$), the hydrogen atom Hamiltonian for a single electron a distance $r$ from a proton of charge $+1$ is

Choose the normalized 1s-type trial function with a free exponent $\zeta$ (not fixed at 1 in advance):

This is a legitimate trial function: it is nodeless, decays to zero at infinity, and is spherically symmetric like the true ground state. The normalization constant is fixed by requiring $\int |\psi|^2\, d^3r = 1$, using the standard result $\int_0^\infty r^2 e^{-2\zeta r}\, dr = 2/(2\zeta)^3 = 1/(4\zeta^3)$, which gives $N^2 \cdot 4\pi \cdot 1/(4\zeta^3) = 1 \Rightarrow N^2 = \zeta^3/\pi$.

Step 2 — Evaluate the kinetic energy expectation value. For a function of the form $e^{-\zeta r}$, the standard result (derivable via the radial Laplacian, or looked up from any table of hydrogenic integrals) is

This comes from writing the radial kinetic energy operator acting on $e^{-\zeta r}$, integrating by parts once so that only first derivatives appear (avoiding the delta-function subtlety at the origin), and using the same radial integral $\int_0^\infty r^2 e^{-2\zeta r}\,dr = 1/(4\zeta^3)$ together with $\int_0^\infty r^{n}e^{-2\zeta r}\,dr = n!/(2\zeta)^{n+1}$ for the resulting terms. The clean scaling $\zeta^2$ is not a coincidence: kinetic energy always scales as the square of an inverse length, and $1/\zeta$ is the natural length scale of this trial function.

Step 3 — Evaluate the potential energy expectation value. The electron-nucleus attraction expectation value is

Using $\int_0^\infty r\,e^{-2\zeta r}\,dr = 1/(2\zeta)^2 = 1/(4\zeta^2)$:

So the attraction scales exactly linearly in $\zeta$, again as expected on dimensional grounds: $\langle 1/r\rangle$ scales as $\zeta$ (inverse length), and the interaction is proportional to that.

Step 4 — Assemble $E(\zeta)$ and minimize. Adding the two pieces:

Differentiate and set to zero:

The second derivative is $d^2E/d\zeta^2 = 1 > 0$, confirming a genuine minimum, not a saddle point or maximum.

Step 5 — Evaluate the optimized energy and compare to the exact answer. Substituting back,

This is exactly the known exact ground-state energy of the hydrogen atom, $E_0 = -\tfrac12$ hartree $= -13.6$ eV, and $\zeta_{\text{opt}} = 1$ reproduces the exact hydrogenic wavefunction $\psi_{1s} = \pi^{-1/2}e^{-r}$. This exact agreement is not an accident of luck — it occurs precisely because the trial family $e^{-\zeta r}$ happens to contain the true ground-state solution as one of its members (at $\zeta = 1$), so the variational search finds the true answer sitting inside the family it searched. This is the essential control experiment: before trusting the method on a problem without an exact answer (helium), you must first confirm it reproduces a known exact answer perfectly when the trial family is rich enough to contain it.

Goal. Use a product trial function of two hydrogen-like 1s orbitals sharing a common variational exponent $\zeta$, include the electron-electron repulsion term exactly, minimize the resulting energy expression, and interpret the optimal $\zeta$ as an effective nuclear charge.

Step 1 — Set up the Hamiltonian and trial function. The helium atom Hamiltonian (nucleus fixed at the origin, charge $Z=2$, electrons labeled 1 and 2) is

The trial wavefunction is a simple orbital product, ignoring for now electron correlation beyond what a shared screening constant can capture:

Both electrons occupy the same orbital with the same exponent $\zeta$; this is the crucial simplifying assumption. Physically $\zeta$ represents an effective charge each electron feels — reduced from the bare $Z=2$ because the other electron's density partially cancels (screens) the nuclear attraction.

Step 2 — Evaluate the one-electron kinetic and nuclear-attraction terms. Because the trial function is a simple product and the Hamiltonian's one-electron operators act on each coordinate separately, the kinetic energy expectation value is just twice the hydrogen-like result derived above:

For the nuclear attraction, note the Hamiltonian has nuclear charge $Z=2$ (not $\zeta$) — the exponent $\zeta$ is a trial parameter, distinct from the actual charge in $\hat H$. Reusing the Step 3 result from the hydrogen derivation, but now with attraction strength $Z=2$ acting on an orbital of exponent $\zeta$:

This is the key place where $\zeta$ (orbital shape) and $Z$ (actual nuclear charge in the Hamiltonian) must be kept distinct — conflating them is the single most common algebra error in this calculation.

Step 3 — Evaluate the electron-electron repulsion integral. This is the genuinely new piece, absent from the hydrogen problem. The two-electron Coulomb repulsion integral for two 1s-type orbitals of common exponent $\zeta$ is a classic result:

Evaluating this six-dimensional integral (by expanding $1/r_{12}$ in Legendre polynomials and integrating the angular parts to zero except the $\ell=0$ term, then doing the remaining double radial integral) gives the closed-form result

This scaling — linear in $\zeta$, with coefficient exactly $5/8$ — is one of the standard results of atomic structure theory and is worth memorizing: for two electrons in the same 1s-type orbital of exponent $\zeta$, the average repulsion energy is $\tfrac58\zeta$ hartree.

Step 4 — Assemble the total energy functional. Collecting kinetic, nuclear attraction, and repulsion terms:

Here the two linear-in-$\zeta$ contributions have been combined: $-4\zeta + \tfrac58\zeta = -\left(4 - \tfrac58\right)\zeta = -\tfrac{27}{8}\zeta$. Already the structure is illuminating: the coefficient $27/8$ is what remains of the bare attraction $2Z=4$ after the repulsion term has clawed back $\tfrac58$ of a unit of charge-times-exponent — a first quantitative glimpse of screening.

Step 5 — Minimize over $\zeta$. Differentiate:

Solve:

The second derivative $d^2E/d\zeta^2 = 2 > 0$ confirms a true minimum. Note immediately that $\zeta_{\text{opt}} = 1.6875 < Z = 2$: the optimal orbital exponent is smaller than the bare nuclear charge, meaning the optimized orbital is more diffuse than a naive unscreened hydrogenic orbital with $\zeta = 2$ would be. Each electron, on average, shields $2 - 1.6875 = 0.3125$ units of nuclear charge from the other — this is the celebrated screening constant of helium's simplest variational treatment.

Step 6 — Evaluate the optimized energy. Substitute $\zeta_{\text{opt}} = 27/16$ back into $E(\zeta) = \zeta^2 - \tfrac{27}{8}\zeta$. It is cleanest to write $E(\zeta_{\text{opt}}) = \zeta_{\text{opt}}\left(\zeta_{\text{opt}} - \tfrac{27}{8}\right)$:

Converting to electron volts using $1$ hartree $= 27.211$ eV gives $E \approx -77.5$ eV. Comparing to the experimentally determined helium ground-state energy of $-2.9037$ hartree ($-79.0$ eV), the simple one-parameter screened variational treatment recovers

of the true binding energy — a remarkably good result from a one-parameter trial function, entirely because the parameter was allowed to relax to its physically meaningful screened value rather than being pinned at the bare nuclear charge.

Example 1 (foundational) — reproduce the hydrogen exponent optimization with explicit numeric bookkeeping. Suppose you did not trust the closed-form kinetic and potential integrals and instead wanted to check the result at a nearby, deliberately non-round trial value, say $\zeta = 0.85$, and confirm the energy lies above the optimum at $\zeta=1$. Using $E(\zeta) = \zeta^2/2 - \zeta$:

Compare to $E(1) = -0.5$ hartree exactly. Indeed $-0.48875 > -0.5$, consistent with the variational theorem: any non-optimal $\zeta$ gives an energy above the true minimum. Try the other side, $\zeta = 1.20$:

Again $-0.48 > -0.5$. Both a smaller and a larger $\zeta$ than the optimum give a worse (higher) energy, confirming the shape of the parabola-like $E(\zeta)$ curve and the location of its unique minimum at $\zeta=1$. This numeric bracket is the standard way to build confidence in a closed-form optimization before trusting it on a harder problem.

Example 2 (applied) — estimate the ionization energy of helium from the screened variational energy. The first ionization energy of helium is defined as the energy to remove one electron, leaving $\text{He}^+$, which is a hydrogen-like ion with $Z=2$ and exact ground-state energy

Using the variational helium ground-state energy from the derivation above, $E(\text{He}) = -2.84765625$ hartree, the estimated first ionization energy is

Converting using $1$ hartree $=27.211$ eV:

The experimental first ionization energy of helium is $24.59$ eV. The variational estimate is low by about $1.5$ eV, or roughly $6\%$ — the error stems entirely from the simple trial function's failure to capture electron correlation (the tendency of electrons to actively avoid each other beyond what a shared mean-field exponent can encode) and from restricting $\text{He}^+$ to its exact value while treating He only approximately, an asymmetric treatment that exaggerates the apparent gap. Nonetheless a one-parameter, back-of-envelope calculation predicting an ionization energy within about 6% of experiment is a genuine success of the method, not a coincidence.

Example 3 (exam / competition level) — a trial function with an independently variable screening constant distinguishing nuclear attraction from orbital shape. A sharper trial function sometimes used treats the screening as its own parameter $s$, writing the orbital exponent as $\zeta = Z - s$ and asking for the optimal screening constant $s$ directly, holding $Z$ fixed at the true nuclear charge. Show that this reparametrization reproduces the same physics as directly optimizing $\zeta$, and find $s_{\text{opt}}$ for helium. Starting from $E(\zeta) = \zeta^2 - \left(2Z - \tfrac58\right)\zeta$ (the general form of the Step 4 result above, generalized to arbitrary $Z$ rather than fixing $Z=2$ immediately — check: setting $Z=2$ recovers $2Z - 5/8 = 4 - 5/8 = 27/8$ as before). Substitute $\zeta = Z-s$:

Differentiate with respect to $s$ (note $d\zeta/ds = -1$, so chain rule flips signs relative to differentiating in $\zeta$ directly):

Solve for $(Z-s)$, i.e. for $\zeta$:

This exactly matches the screening deduced earlier by inspection ($2 - 27/16 = 32/16 - 27/16 = 5/16$), confirming that reparametrizing the same one-parameter family in terms of a screening constant rather than an orbital exponent is merely a change of variable — the physics and the optimum energy are identical, only the labeling of the free parameter differs. This is an important general lesson: the optimized energy from a single-parameter variational search is a property of the trial family, not of how you choose to write the parameter down. For a general two-electron ion of nuclear charge $Z$ (isoelectronic with helium, e.g. $\text{Li}^+$, $\text{Be}^{2+}$), the same algebra gives $\zeta_{\text{opt}} = Z - 5/16$ and

For $Z=3$ (Li$^+$), this predicts $E = -(3 - 0.3125)^2 = -(2.6875)^2 = -7.2227$ hartree, compared to the experimental value of $-7.2799$ hartree — again an error of well under 1%, and the same functional form scales smoothly across the isoelectronic series without any new derivation required.

The idea of optimizing a variational trial function with a floating exponent traces most directly to work in the 1920s and 1930s applying the newly formulated Schrödinger equation to atoms beyond hydrogen. The single individual most associated with the practical machinery of orbital exponents and screening constants is John C. Slater, whose empirical rules for effective nuclear charge (Slater's rules, still taught today) were a direct simplification and generalization of exactly the kind of exponent-optimization calculation carried out above for helium. Slater worked in an era when quantum mechanics was barely a decade old and multi-electron atoms seemed forbiddingly difficult; his insight was that a huge amount of atomic and molecular structure could be captured by treating orbital exponents as adjustable, physically meaningful parameters rather than fixed hydrogenic values, laying groundwork that later matured into the Slater-type orbital basis functions used throughout quantum chemistry.