Diffusion Quantum Monte Carlo Calculations of Excited States of Silicon

Abstract

The band structure of silicon is calculated at the Gamma, X, and L wave vectors using diffusion quantum Monte Carlo methods. Excited states are formed by promoting an electron from the valence band into the conduction band. We obtain good agreement with experiment for states around the gap region and demonstrate that the method works equally well for direct and indirect excitations, and that one can calculate many excited states at each wave vector. This work establishes the fixed-node DMC approach as an accurate method for calculating the energies of low lying excitations in solids. Comment: 5 pages, 1 figure

Full text

Page 1

arXiv:cond-mat/9803207v1 17 Mar 1998
Diffusion Quantum Monte Carlo Calculations of Excited States of Silicon
A. J. Williamson∗, Randolph Q. Hood, R. J. Needs, and G. Rajagopal
Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK
(February 1, 2008)
The band structure of silicon is calculated at the Γ, X, and L wave vectors using diffusion quantum Monte Carlo
methods. Excited states are formed by promoting an electron from the valence band into the conduction band.
We obtain good agreement with experiment for states around the gap region and demonstrate that the method
works equally well for direct and indirect excitations, and that one can calculate many excited states at each wave
vector. This work establishes the fixed-node DMC approach as an accurate method for calculating the energies
of low lying excitations in solids.
PACS: 71.10.-w, 71.20.-b, 71.55.Cn
I. INTRODUCTION
Electronic excitations play a crucial role in the physics
and chemistry of atoms, solids and molecules. Calculat-
ing excitation energies in large systems is a challenge for
theoretical techniques because an accurate description re-
quires a realistic treatment of the electron correlations.
The well-known Hartree-Fock (HF) method includes ex-
change but not correlation effects, and therefore overes-
timates band gaps and band widths by a large amount,
while Kohn-Sham density functional calculations within
the local density approximation (LDA) underestimate
band gaps. Here we report a study of excitation energies
in bulk silicon using the diffusion quantum Monte Carlo
(DMC) method1,2. The DMC method is very promising
for applications to condensed matter because (i) it ex-
plicitly includes electron-electron correlation effects, and
(ii) it scales reasonably well with system size, with the
computational cost increasing as the cube of the number
of electrons.
It is now well established that DMC calculations can
give an excellent description of electron correlations in
the ground state. The range of problems that could be
addressed using DMC would be greatly increased if one
could also obtain accurate excitation energies. Further-
more, the DMC method should be equally applicable to
both strongly and weakly correlated systems. However,
calculating excitation energies in condensed matter sys-
tems is a formidable challenge to DMC techniques be-
cause they are ‘1
N’ effects, i.e., the fractional change in
energy is inversely proportional to the number of elec-
trons in the system.The precision of the calculation
must therefore be sufficient to resolve this energy change
amid the statistical noise. The system must also be large
enough to give a good description of the infinite solid,
which is a severe constraint for small band gap materials
such as our test material, silicon. So far only a few DMC
calculations of excitation energies in solids have been re-
ported.In particular, we note the 8-atom simulation
cell calculations of an energy gap in a molecular nitrogen
solid3and the Γ25′ → X1cand Γ1ν→ X1cexcitations in
carbon diamond4. The excitations in these calculations
were indirect in reciprocal space, so that the excited state
is orthogonal to the ground state by translational sym-
metry. In previous work on solids3–5it has been assumed
that the standard DMC method would give good results
only for the lowest energy state of each symmetry. How-
ever, in this paper we will show that the DMC method
can be applied successfully to a wide range of excitations
in solids.
We have chosen silicon for our study because (i) DMC
gives a good account of the ground state6, (ii) electron
correlations significantly affect the band energies, and
(iii) results from other calculational methods, such as
the LDA, HF7, and GW8–11approximations are well es-
tablished, and a large amount of experimental data also
exists. In this paper we explore the limits of the DMC
method by calculating excitations which are direct and
indirect in reciprocal space, and including several excita-
tions at each wave vector.
It is important to distinguish between different types of
excitation energy. In quasiparticle theory the quasipar-
ticles correspond to the poles of the one-particle Green
function and are equal to the energies for adding an elec-
tron to the system or subtracting one from it. The quasi-
particle energies have both real and imaginary parts, the
latter giving the quasiparticle lifetime. These energies
are measured in photoemission and inverse photoemis-
sion experiments. For the minimum gap the imaginary
part of the quasiparticle energy is zero and the quasipar-
ticle has an infinite lifetime. In this case the quasiparticle
energy gap can be written as Eg= EN+1+EN−1−2EN,
where EN+1, EN−1, and EN are the ground state total
energies of the N+1, N−1 and N electron systems. This
energy gap is accessible within DMC methods, but we do
not consider it here. In an optical absorption experiment
a different process occurs in which an electron is excited
from the valence to the conduction band. In this case
an exciton is formed and the lowest excitation energy is
smaller than Eg by the exciton binding energy. In the
calculations reported here we create excitonic states by
exciting electrons from a valence band state into a con-
duction band state. Although the exciton binding energy
is artificially increased by the finite size of our simula-
1

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tion cell, it is still small (about 0.1 eV) and therefore
our results are comparable with theoretical quasiparticle
energies and experimental photoemission data. We re-
mark that excitonic properties are of significant interest
in their own right and the ability to calculate both elec-
tron addition/subtraction energies and excitonic energies
is a significant advantage of the DMC method.
This paper is organised as follows. In Sec. II we pro-
vide a brief description of our calculational method. Sec-
tion III is a detailed discussion of the results of our study
demonstrating the viability of the DMC method for ac-
curately determining the low lying excitation energies in
solids. We conclude with a summary in Section IV.
II. CALCULATIONAL METHODS
In the DMC method1,2imaginary time evolution of
the Schr¨ odinger equation is used to evolve an ensemble
of 3N-dimensional electronic configurations towards the
ground state. Importance sampling is incorporated via
a guiding wave function, Φ. To make the calculations
tractable we use the fixed node approximation, in which
the nodal surface of the wave function is constrained to
equal that of Φ. The fixed-node DMC method gener-
ates the distribution ΦΨ, where Ψ is the best (lowest
energy) wave function with the same nodes as Φ. The
accuracy of the fixed node approximation can be tested
on small systems and normally leads to very satisfactory
results2. We also use the short-time approximation for
the Green’s function, whose effect can be tested and made
very small. We used a time step of 0.015 a.u., which gives
small time-step errors in silicon6. The average number of
configurations in the ensemble was 384, and between 1220
and 2125 moves of all the electrons in all the configura-
tions were attempted, except for the ground state where
3848 moves were attempted. In all cases the acceptance-
rejection ratio was greater than 99.7%.
We used a fcc simulation cell containing 16 silicon
atoms, employing periodic boundary conditions to reduce
the finite size effects. The Si4+ions were represented by
a norm-conserving non-local LDA pseudopotential, and
the non-local energy was evaluated using the “locality
approximation”12. The non-local potential was sampled
using the techniques of Fahy et al.13Our guiding wave
functions are of the Slater-Jastrow type:
Φ = D↑D↓exp


N
?
i=1
χ(ri) −
N
?
i<j
u(rij)


,
(1)
where there are N electrons in the simulation cell, χ is
a one-body function, u is a two-body correlation factor
which depends on the relative spins of the two electrons,
and D↑and D↓are Slater determinants of up- and down-
spin single-particle orbitals.
expansion for the χ function which was constrained to
We used a Fourier series
have the full symmetry of the diamond structure and con-
tains 6 free parameters. The parallel- and antiparallel-
spin u functions were constrained to obey the cusp con-
siditions14and contained a polynomial part with 11 free
parameters, as described in Ref. 15. The guiding wave
function contained a total of 28 parameters, whose op-
timal values were obtained by minimizing the variance
of the energy using 105statistically independent elec-
tron configurations16,15, which were regenerated several
times during the minimization procedure. The optimal
parameter values were obtained for the ground state wave
function and were used for the excited states as well. Be-
cause the parameters occur only in the nodeless Jastrow
factor this procedure does not bias the DMC excitation
energies, which depend only on the nodal surfaces of the
guiding wave functions.
Our calculations are for many-body Bloch wave func-
tions having definite values of the crystal momentum or
wave vector, k. The Slater determinants were formed
from the LDA orbitals calculated at the Γ-point of the
Brillouin zone of the simulation cell, which unfolds to the
Γ, X and L points of the primitive Brillouin zone17. The
wave vector of a determinant of these orbitals is equal to
one of the Γ, X or L wave vectors. The determinant for
the ground state guiding wave function was constructed
from the valence band orbitals at the Γ, X and L points,
and has k=0. The excited state guiding wave functions
were formed by replacing an orbital in either the up- or
down-spin determinants of the ground state wave func-
tion by a conduction band orbital.
The computational demands of DMC calculations are
such that presently it is not feasible to calculate excita-
tion energies with cells larger than our 16 atom cell, and
therefore it is important to consider the finite size effects.
The finite size effects can be divided into “independent-
particle finite size effects” (IPFSE), which can be mod-
elled by LDA calculations, and “Coulomb finite size
effects” (CFSE), which arise from the explicit use of
Coulomb interactions between the particles. The HF,
LDA and DMC ground state energies for the 16 atom
cell are −103.67, −106.61 and −107.31(1) eV per atom,
respectively. We estimate finite size errors in the ground
state calculations by performing calculations on simu-
lation cells containing 250 atoms (HF and LDA) and
128 atoms (DMC), giving corrections of −0.48, −1.31
and −1.02(3) eV per atom, respectively. The IPFSE are
therefore quite large, being about 21 eV for the 16 atom
cell used here.However, LDA calculations performed
for a number of simulation cell sizes show that the can-
cellation between the errors in the ground and excited
states is so good that the resulting finite size errors in
the LDA excitation energies of the 16 atom cell are less
than 0.1 eV. We expect a similar cancellation of errors in
the DMC cancellations. In our DMC simulations we have
also used a new formulation of the electron-electron inter-
action within periodic boundary conditions18which gives
much smaller CFSE than the standard Ewald form19.
However, the CFSE also tend to cancel in the excitation
2

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energies and we obtained almost identical results using
our new interaction and the Ewald interaction.
Each of the guiding wave functions we use contains
some spin contamination, i.e., they are not eigenstates of
ˆS2but are admixtures of different spin components. We
have investigated the effect of this spin contamination by
calculating the Γ25′ →X1cexcitation energy using a spin
contaminated single determinant and a two-determinant
spin-singlet wave function20. These calculations gave en-
ergies differing by less than the statistical noise of 0.2
eV. The exciton binding energy in our calculations is en-
hanced because the exciton is confined to the simulation
cell. Following Ref. 3 we use the Mott-Wannier formula
to approximate the binding energy of the localized exci-
ton, giving 0.1 eV, which is smaller than the statistical
noise in our calculations. Finally we investigated the ef-
fect on the DMC energy of using single-particle orbitals
which had been relaxed by performing LDA calculations
in the presence of the excitation. The resulting changes
in the excitation energies were smaller than the statistical
noise.
III. RESULTS AND ANALYSIS
In Table I we give our results for the 27 excitations
studied, together with HF, GW, and LDA data. The
characteristics of the HF, GW, and LDA excitation en-
ergies are well known. The GW approximation gives ex-
tremely good excitation energies for weakly correlated
systems such as silicon, while the LDA excitation ener-
gies are too small by 0.7-1.0 eV, and the HF excitation
energies are much too large. The agreement between the
DMC and GW excitation energies is good for the low
energy excitations, but poorer for the higher energy ex-
citations. The percentage, αij, of the correlation energy
retrieved by our DMC calculation for the state formed
by exciting an electron from single-particle orbital i to j
is
αij=Eij
DMC− Eij
Eij
HF+ α0E0
HF+ E0
c
exact− Eij
c
× 100 .
(2)
The values of Eij
estimate the correlation energy for the ground state, E0
to be about -60 eV per simulation cell, and we estimate
the fraction of the ground state correlation energy re-
trieved by the DMC calculation, α0, to be in the range
90-100%. For the purposes of this comparison we use
the GW energies from Ref. 11 for the Eij
the incompleteness and uncertainty of the experimental
data. We note that the various GW calculations for sili-
con9–11are in excellent agreement with one another and
are also in good agreement with the available experimen-
tal data. The experimental data are reported as band en-
ergies, so that to form the excitation energies we have to
take differences between the experimental values, which
adds to the uncertainties. When we present our results
DMCand Eij
HFare given in Table I. We
c,
exactbecause of
for band energies we will compare with the experimental
data, which is given in Table II.
The values of αij given by this analysis slowly de-
crease with increasing excitation energy, so that for the
largest excitation energies the αij are 2-3 percentage
points smaller than for the smallest excitation energies.
This conclusion is insensitive to the values of E0
This suggests the following rationale for our results. The
HF excitation energies are much too large because the
correlation energy is neglected. The fraction of the cor-
relation energy retrieved by our DMC calculations slowly
decreases with increasing excitation energy. However, be-
cause the contribution of the correlation increases rapidly
with increasing excitation energy the DMC excitation en-
ergies are somewhat too large. This analysis indicates
that the residual errors in the DMC excitation energies
are mostly due to the errors in the nodal surfaces of the
excited state guiding wave functions rather than in the
ground state, and that the quality of the nodal surfaces
falls with increasing excitation energy.
To study whether the DMC method works for direct
excitations we considered pairs of excitations where a
single-particle orbital with a particular wave vector is re-
moved from the determinant and replaced (i) by a higher
energy orbital at the same wave vector and (ii) by an
equivalent higher energy orbital at a different wave vec-
tor. For example, when calculating the X4→ X1cexci-
tation energy, we replace an X4 orbital at a particular
X-point by an X1corbital at the same X-point to give
a direct excitation, while for the X4→X∗
replace the X4orbital with an X1corbital from a differ-
ent X-point, giving an indirect excitation. We have in-
vestigated three such direct-indirect excitation pairs and
found reasonable consistency between the results. We
have calculated a total of 6 direct (Γ-point) excitations
and the level of accuracy is indistinguishable from that
for indirect excitations, showing that the DMC method
can be applied to direct excitations. In addition we have
successfully calculated a total of 10 excitation energies at
the X-point and 11 at the L-point, which demonstrates
that the DMC method works for higher excitations as
well.
The 27 entries in Table I correspond to transitions
between 7 valence and 5 conduction band energy lev-
els.To obtain DMC band energies, ǫi, we performed
a least squares fit to the DMC data by minimizing
Σ[Eij
sum is over the 27 excitation energies, Eij
Table I. The resulting band energies are given in Table II
together with other theoretical data and with experimen-
tal data. For greater clarity the DMC band energies are
plotted in Fig. 1 together with an empirical pseudopo-
tential band structure21which is in good agreement with
the available experimental data. The energies at the top
of the valence band have been aligned. The DMC band
energies are very much better than the HF values because
of the inclusion of correlation effects. For the lower part
cand α0.
1cexcitation we
DMC− (ǫi− ǫj)]2with respect to the ǫi, where the
DMC, listed in
3

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of the valence band the DMC energies lie consistently
1 to 1.5 eV below the empirical pseudopotential, GW,
and experimental data. The DMC band energies around
the gap region are in good agreement with the empirical
pseudopotential, GW, and experimental data.
The success of our excitation energy calculations is
very encouraging. To appreciate why our calculations
are so successful we must consider the fixed-node DMC
algorithm in more detail. First we consider ground state
calculations. The fixed-node DMC energy is a variational
bound on the exact ground state energy, and if the nodal
surface of the guiding wave function is exact then the
resulting fixed-node DMC energy is also exact22. The
exact ground state wave function “tiles the configuration
space”, which means that all nodal pockets are related by
permutation symmetry23. Normally one chooses guiding
wave functions which also have the tiling property and
consequently the DMC simulation may be performed in
any subset of the nodal pockets.
For excited states the situation is somewhat different.
If the fixed-node constraint could be removed the algo-
rithm would, in the limit of a long simulation, give the
ground state energy. However, the key point is that the
fixed-node approximation prevents this collapse and al-
lows us to obtain good approximations for excited state
energies. One can readily show that if the nodal surface of
the guiding wave function is exactly that of the nth eigen-
state, then the fixed-node DMC procedure gives the exact
energy of the nth eigenstate, regardless of whether the
guiding wave function overlaps with lower energy states.
However, if we use a guiding wave function that does not
have the exact nodal surface of the excited state we are
modeling, the resulting DMC energy may not be above
the exact energy of that state because of the possibility
of mixing in lower energy states with the same symmetry.
Another complication arises from the fact that the
nodal pockets of approximate excited state wave func-
tions may be inequivalent. Therefore, as the DMC simu-
lation proceeds, the population of configurations in some
nodal pockets will dominate over others, and in principle
the results could depend on which of the pockets were
occupied at the start of the simulation. An illustration
of this type of behaviour for the first excited state of an
electron in an infinite square well is given on page 186 of
Ref. 2, which shows the DMC energy decreasing linearly
with the magnitude of the error in the nodal position.
Although there are significant differences between
ground and excited state fixed-node DMC calculations,
the criterion for obtaining good energies is the same, i.e.,
the nodal surface of the guiding wave function must be
of good quality. Our results demonstrate that the nodal
surfaces from determinants of LDA orbitals are fully ad-
equate for calculating excitation energies around the gap
region in silicon, but give poorer results for higher ex-
citation energies. It will therefore be necessary to opti-
mize the nodal surfaces of the guiding wave functions to
obtain more accurate estimates of the higher excitation
energies. We note that a DMC calculation by Grimes et
al.24for an excited state of the hydrogen molecule with
the same symmetry as the ground state also gave a good
excitation energy. This result lends further support to
our contention that the fixed-node DMC method can be
applied to a wide range of excited states.
IV. SUMMARY
In conclusion, the DMC method is a stable and accu-
rate method for calculating low lying excitation energies
in solids. The fixed-node approximation works to our
advantage by preventing collapse to lower energy states.
The accuracy of the excited state energies is determined
by the quality of the nodal surfaces of the guiding wave
functions. We have obtained good values for the low lying
excitation energies, including direct and indirect excita-
tions, and including several excitations at each wave vec-
tor, which indicates that the nodal surfaces of our guid-
ing wave functions are of good quality. We have demon-
strated that DMC calculations can be used to calculate
direct excitations, which is important because it allows
us to obtain excitation energies when there is no un-
derlying translational symmetry. The fixed-node DMC
method provides a unified framework for calculating ac-
curate ground and excited state energies.
ACKNOWLEDGEMENTS
We thank Matthew Foulkes for helpful conversations.
Financial support was provided by the Engineering and
Physical Sciences Research Council (UK). Our calcula-
tions are performed on the CRAY-T3D at the Edinburgh
Parallel Computing Centre, and the Hitachi SR2201 lo-
cated at the Cambridge HPCF. Guna Rajagopal ac-
knowledges support from Hitachi Ltd.
∗Present address: National Renewable Energy Laboratory,
Golden, Colorado 80401, USA.
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Excitation
Γ1v→Γ2′
Γ1v→Γ15
Γ1v→L1c
Γ1v→X1c
Γ25′ →L3
Γ25′ →Γ15
Γ25′ →L1c
Γ25′ →X1c
X1v→Γ15
X1v→X∗
X4→Γ15
X4→Γ2′
X4→L1c
X4→X1c
X4→X∗
L2′ →L∗
L2′ →Γ15
L1v→L∗
k
DMC
18.05
17.38
16.27
14.93
4.75
3.82
2.35
1.34
12.42
10.37
6.91
7.92
5.71
5.12
4.86
15.60
14.75
12.29
HF7
27.9
26.9
25.4
24.2
8.7
8.0
6.5
5.3
20.5
17.8
12.7
13.7
11.2
10.0
10.0
24.1
23.4
19.8
GW11
15.84
15.31
14.14
13.38
4.25
3.36
2.19
1.43
11.31
9.38
6.29
6.82
5.12
4.36
4.36
13.95
13.06
11.39
LDA
15.14
14.50
13.39
12.58
3.31
2.55
1.44
0.63
10.36
8.44
5.39
6.03
4.28
3.47
3.47
12.92
12.16
10.30
Γ
Γ
L
X
L
Γ
L
X
X
L
X
1c
X
L
Γ
L
X
L
1c
3
3
X
L1v→Γ15
L1v→L1c
L1v→L∗
L3′ →Γ2′
L3′ →L∗
L3′ →Γ15
L3′ →L1c
L3′ →L∗
L3′ →X1c
L
Γ
X
L
11.55
10.80
9.87
6.00
5.76
4.96
3.90
3.82
2.83
19.1
17.6
17.6
11.0
10.7
10.0
8.5
8.5
7.3
10.50
9.33
9.33
5.14
5.50
4.61
3.44
3.44
2.68
9.54
8.43
8.43
4.38
4.50
3.74
2.63
2.63
1.82
1c
3
X
L
Γ
X
L
1c
TABLE I. Excitation energies in eV calculated with the
DMC, HF, GW, and LDA methods. The wave vector of the
excited state wave function is denoted by k. The statistical
error bars on the DMC energies are ±0.2 eV.
Band
Γ25′
Γ15
Γ2′
Γ1
X1c
X4
X1v
L1c
L3
L3′
L1v
L2′
DMCa
0.00
3.70
4.57
-13.58
1.51
-3.35
-8.79
2.51
4.55
-1.32
-7.81
-11.05
HFb
0.0
8.0
9.0
-18.9
5.3
-4.7
-12.5
6.5
8.7
-2.0
-11.1
-15.4
GWc
0.00
3.36
3.89
-11.95
1.43
-2.93
-7.95
2.19
4.25
-1.25
-7.14
-9.70
LDAa
0.00
2.55
3.19
-11.95
0.63
-2.84
-7.81
1.44
3.31
-1.19
-6.99
-9.61
Empd
0.00
3.42
4.10
-12.36
1.17
-2.86
-7.69
2.23
4.34
-1.23
-6.96
-9.55
Experimente
0.00
3.40,3.05
4.23,4.1
-12.5±0.6
1.25
-3.3±0.2,-2.9
2.4±0.15,2.1
4.15±0.1
-1.2±0.2,-1.5
-6.7±0.2
-9.3±0.4
TABLE II. Band energies of silicon in eV. a- This work.
b- Ref.7. c- Ref.11. d- Ref.21. e- From the compilation given
in Ref.11. The statistical error bars on the DMC energies are
±0.2 eV.
5