3. Slater–Jastrow states, cusps, and nodes
Position in the course: Lesson 3 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Slater determinants; local energy
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Separate antisymmetry from positive correlation
A common real trial state is \(\Psi_T=D_\uparrow D_\downarrow e^{J(R)}\), where each determinant handles exchange within a spin sector and a symmetric Jastrow exponent adjusts pair and higher-body correlation. One may write \(J=\sum_{iI}\chi_I(r_{iI})+\sum_{i<j}u_{\sigma_i\sigma_j}(r_{ij})+\cdots\). The positive factor \(e^J\) changes amplitude and often reduces energy variance, but leaves zeros of the determinants unchanged. Multideterminant expansions, optimized orbitals, and backflow coordinates can alter nodes; their extra cost and optimization difficulty must be considered.
2. Derive the electron–nucleus cusp
Near a point nucleus, consider the spherical average of an \(s\)-like nonzero component \(\psi(r)=\psi_0(1+cr+O(r^2))\). Its radial Laplacian is \(\psi''+2\psi'/r=2c\psi_0/r+O(1)\). Kinetic and Coulomb singular terms in \(H\psi/\psi\) are therefore \(-c/r-Z/r\). Cancellation requires \(c=-Z\), giving
This statement concerns the appropriate spherical component and a Coulomb point nucleus; pseudopotentials remove or replace the singularity and need not satisfy the same cusp. A Gaussian orbital by itself has zero radial slope at the origin and therefore lacks the Coulomb cusp.
3. Electron-pair cusp and spin dependence
For two equal-mass electrons, the relative kinetic operator is \(-\nabla_r^2\) in atomic units, not \(-\nabla_r^2/2\). For an opposite-spin nonzero spherical component \(\psi\approx\psi_0(1+cr)\), singular local energy is \(-2c/r+1/r\), so \(c=1/2\). For same-spin electrons the spatial state vanishes linearly with separation; extracting the \(l=1\) factor leads to cusp coefficient \(1/4\) for the remaining radial amplitude. These coefficients depend on the sign convention: here \(\Psi=e^J D\) and \(u'(0)\) is positive.
4. Worked hydrogen check
For \(\psi=e^{-ar}\) with \(H=-\nabla^2/2-1/r\), \(\nabla^2\psi=(a^2-2a/r)\psi\). Hence \(E_L=-a^2/2+(a-1)/r\). At \(a=1\) the singularity cancels and energy is the constant \(-1/2\). At any other \(a\), the residual \(1/r\) term signals a cusp mismatch. This analytical benchmark checks both the Laplacian and the potential sign.
Cusps concern short-range behavior; they do not ensure correct long-range screening or nodal topology. Correlation functions also need suitable cutoffs, smooth derivatives, and periodic compatibility. A discontinuous derivative can create an artificial kinetic feature. Test the combined logarithmic gradient and Laplacian because \(|\nabla\ln\Psi|^2\) contains cross terms. Checking components separately is necessary but not sufficient for the complete wavefunction.
5. Exercises and limitations
Exercise: Can a real positive Jastrow factor repair a poor fixed-node boundary?
Solution
It cannot move the zeros when finite and positive. It can improve amplitudes, variance, and some pseudopotential behavior, but nodal improvement needs parameters that change the antisymmetric part or its coordinates.
Exercise: Why should one not impose a nuclear Coulomb cusp on a smooth pseudopotential orbital automatically?
Solution
The derivation used a \(-Z/r\) singularity that the pseudopotential model does not contain. Imposing an unrelated singular derivative can worsen the model wavefunction and local energy.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
- Kato (1957), On the eigenfunctions of many-particle systems in quantum mechanics
- Drummond et al. (2004), Jastrow correlation factor for atoms, molecules, and solids
Related theory: Classical Monte Carlo · Molecular methods · Density functional theory
Software connection: CP2K · Quantum ESPRESSO · Gaussian
These software courses provide related background on energies, orbitals, or convergence management; they do not imply that the Monte Carlo or QMC examples on this page were executed there.
7. Related theory and practice
Quantum mechanics · Molecular methods · Monte Carlo · Molecular dynamics · QMCPACK