DMC Results

All of the possible excitations accessible in an n=2, 16 atom simulation cell are illustrated in figure . The promotion of an electron from the orbital representing the top of the valence band at the -point to the orbital representing the bottom of the conduction band at the X-point ( ) was used as a test bed for the following variations in the DMC technique:

1. As described in section , the use of a single product of Slater determinants for up and down spin electrons in which either the up-spin determinant contains an excited state orbital and the down-spin determinant contains ground state orbitals or vice versa, is spin contaminated. There are no calculations in the literature to indicate how severe the effect of this contamination is. Using the dual determinant wavefunction given in Eq.() slows down the code by almost a factor of two (as operations on the determinant(s) dominate the calculation), so it was decided to perform tests using single and dual determinants to determine whether the effect of the spin contamination is resolvable from the statistical noise and therefore whether the use of dual determinant wavefunctions is necessary.
2. Separate DMC calculations were performed using the two electron-electron interactions for the excited state described in Eqs.() and (). These correspond to using either the ground state or excited state LDA charge density in the Hamiltonian of Eq.().
3. The one-electron orbitals used in the Slater determinant part of the trial/guiding wavefunction are generated from an LDA calculation (see chapter ). Within the LDA there is only a very small change in the excitation energy if one relaxes the orbitals after promoting an electron to the conduction band compared with using fixed ground state orbitals to calculate all energy differences. Two separate DMC calculations were performed on the excited state to check whether using relaxed rather than fixed LDA orbitals in the Slater determinant had a significant effect on the DMC excitation energy. This could happen if the process of relaxing the LDA orbitals significantly altered the nodal structure produced by the Slater determinant.

The following DMC calculations were performed:

1. A wavefunction containing a full dual determinant, as shown in Eq.(), was used to represent the state in which one electron is promoted from the top of the valence band to the bottom of the conduction band. The LDA orbitals were not relaxed from their ground state forms. The enhanced interaction of Eq.() was used to model the electron-electron interaction. The total energy for the excited state system was -107.21 0.05 eV per atom.
2. A wavefunction containing a single determinant of one-electron orbitals was used. Again the enhanced interaction of Eq.() was used to model the electron-electron interaction. Again the LDA orbitals were not relaxed from their ground state forms. The total energy for the excited state system was -107.22 0.02 eV per atom.
3. A wavefunction containing a single determinant of one-electron orbitals was used again. This time the less sophisticated interaction of Eq.() was used to model the electron-electron interaction and the one-electron orbitals were relaxed in the LDA. The total energy for the excited state system was -107.22 0.02 eV per atom.

The above results suggest that for the test case of promoting a single electron from the top of the valence band at the -point to the bottom of the conduction band at the X-point, (i) The effect of spin contamination is not resolvable from the statistical noise, (ii) The two choices of electron-electron interaction yield the same energy, and (iii) the effect of relaxing the one-electron orbitals within the LDA has no significant effect on the total energy.

In the light of the above results, all the following promotion calculations are based on the enhanced interaction of Eq.(). Although this is the more sophisticated interaction, it is actually slightly simpler to implement, because it only relies on the LDA ground state charge density as an input, whereas the less sophisticated interaction of Eq.() requires a separate LDA calculation of each excited state charge density for use as an input. Also, in all the following calculations, the same LDA orbitals obtained from a ground state calculation, have been used to construct the Slater determinant, again to simplify the setup procedure. A single determinantal product was used to represent the excited state to speed up the computation. The DMC calculations were performed using 384 configurations distributed over 128 nodes of the parallel computer. The diffusion algorithm used between 1500 and 2000 time steps. Approximately 250 of these time steps were required for the initial propagation stage of the algorithm (see chapter ) and the remainder were used to accumulate statistics.

The results are shown in table . Again, the equivalent n=2 HF and LDA results have been included for comparison. It should be noted that is in the addition and subtraction of electron results, the LDA results contain only a small finite size effect, whereas the HF results contain a large finite size effect.

On the whole, the calculations appear extremely successful, with a significant fraction of the results in agreement with experiment to within error bars. Those calculations which significantly disagree with experiment all overestimate the size of the gap. This would be consistent with the quality of the trial wavefunction for the excited state is not being as good as that for the ground state. In particular the nodal structure of the excited states may not resemble the true nodal structure as closely as that of the ground states. These approximations to the excited states will tend to increase the estimate of the energy of the excited state and hence produce estimates of the gap that are too large.

  
Figure: Pseudopotential band structure of silicon showing the , X and L-points, taken from Ref.[94]. All possible excitations from the top of the valence band to the bottom of the conduction band are shown. Excitations from the -point are shown in blue, from the X-point in red and from the L-point in green.

 

Promotion DMC Gap (eV) Expt. Gap (eV)[94] LDA Gap (eV) HF Gap (eV)

1.2 0.46 2.73

6.3 5.36 8.81

2.4 1.39 4.00

4.6 3.66 6.36

5.2 4.32 7.31

2.4 1.69 4.09

4.1 3.39 6.04

3.4 2.62 5.36

 

The average deviation of the DMC energies from experiment is 0.3 eV. For the LDA it is -1.0 eV and for HF it is +6.8 eV.

• Calculating Band Widths
• Excitonic Effects