Nodeless ground state wavefunctions

The procedures described above for DQMC and GFQMC lead to the lowest energy solutions for boson systems that are nodeless ground state wave-functions. They also lead to the ground state in the case of two electrons (fermions) of opposite spin for which the wavefunction is symmetric to the exchange of the two electrons. For a system of two or more electrons of the same spin, the wavefunction must be antisymmetric to the exchange of electrons of the same spin and must contain one or nodal hypersurfaces. The treatment of systems with nodes requires that the solutions be constrained to the appropriate antisymmetry. 

Since this function is nodeless, we identify it with the ground state of the hydrogenlike atom. Multipyling (7.35) by the spherical harmonic Too = l/ /4jr, we obtain the complete wavefunction (7.27), which in this case reduces to... 

The n5-orbitals are all spherically symmetrical, being associated with a constant angular factor, the spherical harmonic Too = 1 /V4. They have n — radial nodes—spherical shells on which the wavefunction equals zero. The I5 ground state is nodeless, and the number of nodes increases with energy, in a pattern now familiar from our study of the pai1icle-in-a-box and harmonic oscillator. The 2s orbital, willi its radial node at r = 2 bohr. is alsit shown in Fig. 7.3. 

The velocity diverges at the nodes and for the Coulomb potential has a discontinuity at particle coincidences both for approximate and for the true wavefunction. For the nodeless wavefunction of Lennard-Jones particles in their ground state [5], V diverges as r 6 in the interparticle distance r. Since the only other problem for these bosonic systems is the divergence of l at particle coincidences, we continue our discussion for electronic systems and refer the interested reader to the literature [5] for details. 

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