HAL Id: jpa-00215060
https://hal.archives-ouvertes.fr/jpa-00215060
Submitted on 1 Jan 1972
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
AN “ ORBITAL CORRECTION METHOD ” FOR SOLID-STATE CALCULATION
W. Harrison
To cite this version:
W. Harrison. AN “ ORBITAL CORRECTION METHOD ” FOR SOLID-STATE CALCULATION.
Journal de Physique Colloques, 1972, 33 (C3), pp.C3-179-C3-183. �10.1051/jphyscol:1972326�. �jpa-
00215060�
AN ORBITAL CORRECTION METHOD >>
FOR SOLID-STATE CALCULATIONS (*) W.
A. HARRISONApplied Physics Department, Stanford, California 94305
RhumC. - Un procede de calcul des valeurs propres monoelectroniques et des energies totales dans des systkmes arbitraires a Bte dkvelopp6. On se donne un &tat initial approche et la valeur propre est developpee suivant la cr correction orbitale P, difference entre la vraie fonction propre et la fonction approchk. Si la fonction initiale est fonction propre de quelque hamiltonien de depart, cette methode se reduit B la theorie ordinaire des perturbations, mais ceci n'est jamais un bon choix dans les solides. Dans un metal simple, I'estimation initiale peut 6tre une onde plane unique, orthogonalis6e
A
tous les Btats du cceur : ceci conduit directement B la theorie du pseudo- potentiel. Dans un metal de transition, l'etat initial appropri6 peut &tre une OPW completQ par une combinaison linkaire #&tats atomiques d et ceci conduit aux pseudo-potentiels des mktaux de transition. Dans les cristaux covalents, molCcules ou systemes desordonnks, l'etat de depart peut 6tre semblable B une combinaison lineaire d'orbitales atomiques et la theorie conduit a un dkveloppement systematique en fonction de I'erreur de la methode LCAO qui peut en general&re pousse au second ordre. Le procede est donc plus prkis que la methode LCAO et peut aussi 6tre beaucoup plus simple parce que l'elimination systematique des termes d'ordre superieur dis- pense de I'habituelle diagonalisation de l'hamiltonien. Des applications preliminaires eclairant le problkme des matkriaux covalents d8sordonnes sont discuttes.
Abstract. - A procedure has been developed for computing one-electron eigenvalues and total energies in arbitrary systems. An approximate starting state is written and the eigenvalue is systema- tically expanded in the <( orbital correction )>, the difference between the true eigenstate and the approximation. If the starting approximation is the eigenstate of some starting Hamiltonian, this becomes ordinary perturbation theory, but that is never a good choice in solid-state systems. In a simple metal the initial guess may be a single plane wave, orthogonalized to all core states ; this then leads directly to pseudopotential theory. In a transition metal the appropriate starting state may be an OPW plus a linear combination of atomic d-states and this leads to transition metal pseudopotentials. In covalent crystals, molecules, or disordered systems the starting state may be something akin to a linear combination of atomic orbitals and the theory becomes a systematic expansion in the error in the LCAO method ; it can ordinarily be carried to second order. Thus it is more accurate than LCAO theory but may also be much simpler because the systematic elimi- nation of higher order terms makes the usual diagonalization of the Hamiltonian matrix unneces- sary. Preliminary applications which shed light on the problem of disordered covalent materials are discussed.
I.
Introduction. - The approach t o electronic structure whichI
would like to discuss is perhaps more of a state of mind than a specific procedure.Basically, the idea is to use physical intuition t o esta- blish a good starting approximation t o the one-elec- tron eigenstates and t o systematically make correc- tions. Ordinary perturbation theory differs from this in that the starting approximation is based upon an exactly soluble Hamiltonian. These are in fact so scarce that in most problems there is only one real choice of zero-order Hamiltonian and it becomes difficult t o make full use of what intuition we have.
I n the orbital correction method we are free t o use
(*) Supported in part by the National Science Foundation and in part by the Advance Research Projects Agency through the Center for Materials Research at Stanford University.
our intuition in generating an approximate eigenstate and we expand in the error. We will see that a special case of this is the pseudopotential method in simple metals. There we take as an approximate state a plane wave orthogonalized t o every core state and expand in the c< orbital correction )) t o this orthogonalized plane wave.
There are peculiarities which are common to most applications of this method. First, we are ordinarily limited in the order t o which we can carry the expan- sion. Thus, if our calculation does not suffice to cla- rify a property of interest, it is ordinarily not possible simply to carry the calculation to higher order but we need instead t o improve upon our starting approxi- mation t o the wavefunction. Second, there tends to be an arbitrariness in the expansion, corresponding to the familiar arbitrariness of the pseudopotential
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972326
C3-180 W. A. HARRISON
method. This may be understood by imagining an
orbital correction which when added to an approxi- mate eigenstate gives us the exact (unnormalized) eigenstate. If we add to that orbital correction any multiple of the true eigenstate we again have an orbital correction giving us an exact (unnormalized) eigenstate. Ordinarily, we would choose that multiple so that the total orbital correction is as small as pos- sible, but there are different ways of prescribing such procedures. In the case of simple metals, one proce- dure corresponds to the optimized pseudopotential.
My principal efforts with this method recently have concerned the electronic structure of molecules. Most of the examples I will give will therefore concern molecules. However, it will be apparent that an impor- tant feature of the method is that it is not restricted to small systems but is equally applicable to solids.
We will focus attention particularly upon the total energy of the system and its dependence upon the arrangement of constituent atoms
;this is the problem of chemical bonding. It will be clear, however, that the same approach can shed light on the electronic properties as well. The calculations will in all cases be based upon two important approximations. The first is the inclusion of electron-electron interactions within a self-consistent-field approximation. Thus in all cases we will be doing a one-electron problem based upon a potential which should be self-consis- tently determined. The second approximation is the carrying of the expansion in the orbital correction only to second order. This approximation will entail the discarding of a term in the potential, a step which seems very well justified in the pseudopotential method but less so in molecular problems. However, Maarten Heyn [I] has shown that the discarding of this term corresponds to doing only the first iteration in a pro- cedure which would ultimately iterate to the exact answer. Although this provides some justification for the expansion we are using, it is not possible to carry it to higher order and the procedure must be evaluat- ed by comparison with exact answers or with expe- riments.
11.
The Formulation.- We are seeking the eigen- states of a Hamiltonian
X which contains a kineticenergy operator T and a one-electron potential V.
Thus we wish to solve an eigenvalue equation
We write the true eigenstate I
$> as the sum of our initial guess 1
$,> and an orbital correction I x >.
It will be convenient to normalize the starting wave- function and therefore the true wavefunction is not normalized. We substitute this form in eq. (I), mul- tiply on the left by <
$,I and solve for the energy to obtain
We obtain x as an expansion in plane waves (or other eigenstates of the kinetic energy operator)
;that is,
I x >
= Zq1
q> <
qI x >. We obtain the coeffi- cients by again substituting eq. (2) into eq. (1) but this time multiplying on the left by a plane wave < q I.
We obtain
We regard V I x > as of higher order than I x >
;this is the approximation mentioned in the introduc- tion. We may alternatively say that we use the arbi- trariness of I x > to make a choice such that V I x >
is small. Then substituting eq. (4) in eq. (3), we obtain
where we have now introduced an index i to enume- rate the eigenstates. tji is again the zero-order starting state previously written, \I/o. The only approximation upon the initial problem, eq. (I), has been the drop- ping of V I x > as of higher order. The sum in eq. (5) is formally a second order term and therefore Ei may be replaced by its zero order value <
$iI X I \I/, >
everywhere in that term and the equation for the energy becomes explicit.
We may clarify the meaning of eq. (5) by applying it to the simple metals, where I $ i > becomes a plane wave orthogonalized to every core state I a >. To
show that this is equivalent to a perturbation expan- sion in the pseudopotential requires some algebra and the exact result obtained depends on precisely which terms are regarded as of higher order. Suffice it to say that
X- E operating upon an OPW gives
The first term is a constant depending upon the zero of energy and is not of interest. The second term is precisely the pseudopotential of Phillips and Klein- man [2] operating upon the plane wave 1 k >. In this context we may note also that the final term of eq. (4) which we discarded earlier is indeed of higher order in the pseudopotential. The orbital correction may be taken orthogonal to the core states
;then V I x > becomes precisely the pseudopotential times the orbital correction and is therefore of higher order than the first term which is the lowest order term in the orbital correction.
In the simple metals single orthogonalized plane
waves are indeed appropriate starting points and the
pseudopotential expansion gives us not only a method
of calculating the total energy (and therefore proper- ties such as the vibration spectrum) but also enables us to address the entire spectrum of electronic pro- perties. In a semiconductor, and even more clearly in a disordered semiconductor or molecule, such an OPW is not a suitable starting eigenstate. A suit- able linear combination of atomic orbital would seem much more appropriate. In the treatment of noble metals and transition metals it is necessary to simultaneously work with both types of starting states, OPW's for s-like states and LCAO's for d-like states.
These are the most difficult systems to treat. The cor- responding TM$ (transition metal pseudopotential) method has in fact been carried through [3] but will not be discussed in any detail here. We will discuss principally the orbital correction method as applied to semiconductors and molecules, with particular emphasis on the bonding properties.
We begin with the $i of eq. (5) written as linear combinations of atomic orbitals. To obtain the total energy we must sum over these linear combinations.
It will greatly simplify the problem if the number of states in the basis set in which we expand is equal to the number of linear combinations which we sum
;this basis set might be called the set of bond orbitals.
It is even smaller than what is called a minimum basis set since we discarded at the outset any combinations (ordinarily antibonding combinations) which are not to be occupied. That distinction is not too important since they would be discarded at the end in any case.
However, even the minimum basis set is frequently not adequate in the usual LCAO method since it does not allow for any improvement in the starting wavefunctions. Such improvement in the ordinary LCAO method is obtained only by adding additional atomic states. However, here it is accomplished by the orbital correction and a minimum set or just the basis set of bond orbitals should ordinarily suffice.
This will require some care in the choice of basis since almost always more atomic orbitals than states are involved and bonding combinations of atomic orbitals will be required at the outset. In crystalline silicon for example we will find it appropriate to start with bond- ing combinations of s-p hybrids in each bond
;the errors in this choice are then corrected for by the orbi- tal correction.
With such a basis set of bond orbitals the starting eigenstates are written
where the I a > are the bond orbitals and A , is a square matrix.
If the Ai, formed a unitary matrix and we required only the sum over the first term in eq. (5), that sum could be performed immediately using the unitarity condition and each I
$ i> would be replaced by an
I
a>. The total energy would not depend upon the coefficients A , and we could work directly with our
basis states. This is of course just a statement that the trace of a matrix is unchanged by a unitary trans- formation.
Such a simplification would be of extreme value if it could be made since it would mean that the total energy could be evaluated without the diagonaliza- tion of a matrix of rank equal to the number of elec- trons involved. It is this diagonalization which limits most methods to small molecules. The elimination of this diagonalization would allow us to treat one portion of a large system, such as a typical region within an amorphous semiconductor. It would also enable us to treat a very large molecule, with the effort being only linear in the size of the system.
Unfortunately, in most realistic situations, the matrix A , is not unitary. Even if it were, the depen- dence of the second term in eq. (5) upon E, would prevent the use of the unitarity condition to eliminate the A,. The total energy does depend upon taking the appropriate linear combination. It is possible, however, to use the condition that the A , diagonalize the Hamiltonian matrix to first order, in place of the unitary condition, to evaluate the corresponding sums to second order. Thus it is possible to evaluate the total energy without explicitly performing a diagona- lization. The same simplification carries over to the TM$ method, at least when applied to the noble metals, where it is unnecessary to specify the appro- priate linear combination of d-states and the total energy can be evaluated directly.
111. Applications. - As we have indicated, the orbital correction method becomes the pseudopoten- tial method when applied to simple metals. The appli- cations to disordered as well as,ordered systems are already familiar. In particular the treatment of liquid metals by perturbation theory is standard. Orthogo- nalizing to the core states eliminates all deep bound states from the one-electron problem. The system is sufficiently uniform that any remaining bound states are very shallow in comparison to the Fermi energy and it is quite appropriate to treat the pseudopoten- tial as a perturbation in considering either the resis- tive scattering by the system or the interactions bet-
ween ions.
The application to semiconducting systems has unfortunately just begun. However, the studies of molecular systems which we are pursuing have shed light on the solid-state problem and those aspects of the molecular study may be reviewed.
A study of the central hydride series, methane,
ammonia, water, and hydrofluoric acid has been
undertaken by Richard Meserve
[4].As in the appli-
cation to solid state systems, the first step is the choice
of basis functions in which to expand the starting
wavefunction. The simpler this choice, the simpler
the calculation. We might imagine that as a first
approximation the electron from each hydrogen atom
is donated to the central atom in the central hydride.
C3-182 W. A. HARRISON
Thus, we might take as starting states in the case of
methane the full set of carbon
1s, 2 s, and 2 p orbitals.
These become the bond orbitals and have the added feature that they are orthogonal. (Because they are orthogonal, use of any orthogonal combination of them, such as s-p hybrids, would be mathematically equivalent.) The correction to these orbitals due to the four protons is included in the orbital correction.
Thus our first attempt was the use of Herman-Skill- man [5] carbon orbitals. However, using the corres- ponding methane Hamiltonian the first term in eq. (5) was found to be positive
;i. e., not even bound. Thus the choice looked too poor to be useful and an impro- vement seemed necessary.
Hoping not to lose the great simplicity of these orthogonal orbitals, we sought atomic states computed in the spherical average of the methane Hamiltonian.
Thus the protons were replaced by a spherical globe of charge with radius equal to the interatomic distance and spherical methane orbitals were calculated using a slightly modified Herman-Skillman program [6].
The calculation is essentially a neon-configuration calculation but differs from neon in that four proton charges are distributed on the globe rather than in the nucleus. The total energy of methane based upon these orbitals was remarkably accurate even with the omission of the orbital correction and therefore the second-order term of eq. (5). It leads in particular to the correct proton configuration, and reasonably accurate equilibrium spacing, cohesive energy, and vibrational frequencies. (It is clear however that the correct configuration for ammonia and water cannot be obtained if only the leading term is retained.)
These findings on methane suggest that it is impor- tant not to use atomic orbitals as a basis for the orbital correction method but atomic-like orbitals which are more appropriate to the aggregate system.
It is common in fact in LCAO calculations to use Slater orbitals, adjusting the exponents to minimize the energy. We prefer to use tabulated orbitals and can therefore generalize the methane approach to solids simply by seeking the spherical average of the solid-state potential around each atomic center. It seems likely that a good approximation to this poten- tial would be obtained by including sufficient charge on a globe at the interatomic distance so that a neutral- rare-gas-atom-configuration calculation can be carried out as in methane.
Let us now think in terms of a total energy calcu- lation for a diamond crystal, having computed the appropriate carbon orbitals. This is too large a basis set since there are eight orbitals (including spin dege- neracy) per atom but only four electrons per atom.
It would seem most plausible then to construct bond- ing combinations of s-p-hybrids which would cor- respond to orbitals localized in the bonds. These become our basis of bond orbitals and we could proceed from there. Such calculations have not been carried far enough that we may give any numerical
results but two aspects are quite clear
;both concern the zero-order problem upon which the orbital cor- rection is to be applied. Without orbital corrections the bands become tight binding bands of bonding orbitals. These are just the bands calculated by Hall [7]
assuming overlap only between bond orbitals on adja- cent bonds. This in fact provides a very reasonable starting point for the valence band and there will be corrections both from the more distant overlaps and from the orbital correction. That is gratifying, though we are not proposing the scheme as a band calculation technique. The second feature concerns long-range interactions in the total-energy calculation.
Having transformed the calculation such that all orbitals in eq. (5) are local atomic-like orbitals any long-range interactions must arise formally from long- range Coulomb potentials. Such potentials do arise from the inter-nuclear interactions and in the absence of screening would in fact lead to finite frequencies for long wavelength longitudinal vibrations
;that is, to an ion plasma frequency rather than longitudinal sound. However, there are also zero-order charge distributions due to the bond orbitals which move as the atoms move and in fact provide that aspect of the screening which reinstates the acoustical modes.
These moving electronic charges play the role of the bond charge used by Martin [8] in the calculation of the silicon vibration spectrum. A recalculation of this silicon spectrum, including orbital correction terms, is currently under way
[9].It is not clear in detail how one should proceed with disordered structures. Hopefully, this will become clear as we gain experience with the perfect semi- conductor. It does seem likely that an approach which at the outset separates the bonding and anti-bonding states has much more promise than an approach based upon propagating states which seeks to produce a gap through diffraction.
V.
Summary.- The orbital correction method which we propose may be regarded as a generaliza- tion of the pseudopotential method. The application which we are most heavily pursuing concerns the total energy of molecules. The calculation is carried out in a self-consistent-field approximation and has a starting point closely related to linear combinations of atomic orbitals. The goal is the calculation of the total energy. For that problem it has been possible to transform the system to an entirely local basis and therefore to treat small regions of indefinitely large systems. For this reason it seems a particularly pro- mising approach for disordered systems.
One might be concerned that the starting point
for any particular system might be chosen differently
by different workers and the results are not as unique
as in simple metals where most people would agree
that single orthogonalized plane waves provide a
sound starting point. This flexibility, however, cor-
responds more closely to a choice of models than to
a system with many adjustable parameters. Once an of the reliability of the method. However, it is reaso- Ansatz has been made for the starting point the results nable to expect that, as with the simple metal pseudo- follow uniquely. An insufficient number of calculations potentials, the reliability of the results will vary from have been made to date to allow any real assessment property to property.
References
[I]
HEYN (M.), unpublished.
[6]The author is indebted to Dr. Frank Herman for
[2]PHILLIPS (J. C.) and KLEINMAN
(L.), Phys. Rev., 1959,providing
acopy of the program used in the
116, 287.
Atomic Structure Calculations.
[3] HARRISON (W. A,), Phys. Rev., 1969,181, 1036. [7]
HALL (G. G.),
Phys. Rev., 1953, 90, 317.[4]
MESERVE (R.), unpublished.
[5]
HERMAN
(F.)and SKILLMAN (S.),
Atomic Structure(R.
M.), Phys- 19699 lg6, 871.Calculations
(Prentice-Hall, Inc., Englewood
Clifts [9]CIRACI
(S.),unpublished.
N.
