J. Phys. C : Solid State Phys., 14 (1981) 3795-3806. Printed in Great Britain

A method of embedding

J E Inglesfield Science Research Council, Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK

Received 7 April 1981

Abstract. A surface potential is derived which can be added to the Schrodinger equation for a limited region of space, I, to embed it into a substrate. This potential, which is energy- dependent and non-local, can be found from the Green function for the bulk substrate. The results are based on a variational principle which gives the energy of a state in terms of the wavefunction in I . The embedded Schrodinger equation can be solved by a basis set expan- sion, for the wavefunctions of discrete states and the Green function in the continuum, and the method is demonstrated for the case of a square well.

1. Introduction

When an impurity is introduced into a crystal, or a surface is made, the perturbing potential and the change in charge density are quite localised. This makes it an attractive proposition to solve the Schrodinger equation self-consistently in a cluster of atoms around the impurity, or in the top few surface layers, if possible matching the wave- functions onto the solutions of the Schrodinger equation in the substrate crystal. Most present-day impurity and surface calculations consider the cluster or surface layers in isolation (Ellis et a1 1979, Posternak et al1980), dealing with a sufficiently large number of atoms so that the finite system is representative of an impurity in an infinite crystal, or the surface of a semi-infinite crystal. However, this gives rise to very large computer programs, and it becomes difficult to concentrate on those features in the electronic structure specifically associated with the impurity or surface. It is, in fact, possible to deal with a small Hamiltonian confined to the perturbed region of the crystal, adding on an extra effective potential which automatically ensures that the wavefunctions match onto the substrate-this is embedding.

In our new embedding technique we consider a region around the impurity or the surface layers, region I, to be joined onto the substrate, region 11, over surface S (figure 1). In § 2 we show how a surface potential on S can be added on to the Hamiltonian for I so that solving the Schrodinger equation in this limited region of space gives wave- functions which match correctly onto the substrate. The surface potential is constructed from the Green function for the perfect substrate, and is energy-dependent and complex in the substrate continuum: this broadens the initially discrete states of the finite system I into the continuum. It can be found once and for all for any particular substrate, so that we can proceed to self-consistency within region I , for example, with the same embedding potential. Unlike previous embedding approaches (Grimley and Pisani 1974), any technique can be used to solve the Schrodinger equation in I: by adding on the surface

3795 0022-3719/81/263795 + 12 $01.50 @ 1981 The Institute of Physics

3796 J E Inglesfield

Figure 1 . Region I to be embedded in I1

potential, any existing cluster or layer program can be readily modified to embed it into the substrate.

The usual ways of treating impurity and surface problems are based on Green functions. The effect of a perturbing potential dV(r), associated with an impurity for example, can be found by solving the Lippmann-Schwinger equation, or Dyson's equation:

G(r , r ' ) = Go(r, r ' ) - d3J'Go(r, r")GV(J')G(J', r ' ) , (1) I where CO, G are the Green functions for the unperturbed and perturbed systems. This has been widely used in impurity (Baraff and Schluter 1978, Bernholc et 011978. Zeller and Dederichs 1979), and adsorbate problems (Gunnarsson and Hjelmberg 1975, Lang and Williams 1978), where W ( r ) is localised. It is not suitable for cases where 6V(r) is extended-for example when a surface is created, 6V extends over a half space. An alternative technique is the matching Green function method (Garcia-Moliner and Rubio 1969, Inglesfield 1971), in which the Green function for region I joined on to 11, as in figure 1, is constructed from Green functions for I and I1 separately. This can be applied to a wide range of surface (Inglesfield 1978) and impurity problems (Inglesfield 1972), but unfortunately it involves complicated manipulations of the Green functions over surface S.

Our new method has none of these drawbacks, and to show how it works we apply it here to a square well embedded in a constant potential, considering bound states in § 3 and continuum states in § 4. In subsequent papers we shall embed isolated impurities and clusters of atoms into a metal.

2. The embedding method

We wish to solve the Schrodinger equation in region I (figure l), with extra conditions so that the solutions match correctly onto the substratz 11. To do this we find the expectation value of the energy of a wavefunction @(r) , defined in I as a trial function q(r) , and in I1 as the exact solution of the Schrodinger equation at some energy E , q ( r ) , which matches in amplitude onto q over St:

(-&V? + V(r) - E)v(r) = o r in I1 ( 2 )

vks) = d r d . (3)

t We use atomic units with e = f i = m = 1

A method of embedding 3797

The expectation value is then:

E = 1 d3r@*(r) H @ ( r ) / j d3r@*(r)@(r)

J1d3rq*(r)Hq(r) + .5J11d3rV*(r)~(r) + kfsd2rsq*(rs) (m - - - - ans aV(rs)) ans J d 3 r ~ * ( r ) d r ) + Jrrd3rV*(4W(r)

(4)

where H i s the Hamiltonian and the final surface integral in the numerator comes from the discontinuity in derivative between pl and V.

We now express the derivative dq(rs)/ans in terms of q(rs ) , by using the Green function Go for region 11, satisfying:

( 5 ) ( - l V f + V ( r ) - &)GO(T, r ’ ) = 6(r - r ’ ) r , r‘ in 11.

Multiplying (2) by Go, ( 5 ) by q and integrating through I1 we obtain:

and then from Green’s theorem:

If we construct Go to have zero derivative on S, this gives us an equation relating the amplitude of the wavefunction in I1 and its normal derivative on S:

and putting r on S:

The inverse of this equation gives us what we require - dqians in terms of yt(rs), which we have set equal to q(rs):

(10) a v(rs> - = -2 I d2riG;’(rs, ri)v(ri) .

S

It is in fact quite easy to construct the surface operator Go’ from a surface matrix representation of any Green function for the substrate, using the matching Green function method (Inglesfield 1978).

Substituting (IO) into (4) gives:

apl(rS) E = (1 d3rpl*(r)Hq(r) + E d3rq*(r )v(r ) + 1 d2rgp*(rS) -

-we are still left with the volume integral of ]VI2 over region 11, but this can also be

3798 J E Inglesfield

reduced to a surface integral. Varying E in ( 2 ) , but still ensuring that matches onto q

This expression gives us E purely in terms of a trial function q(r) defined in I and on S : it is a genuine variational principle, giving an upper limit to the energy, and by minimising E with respect to variations in q, and energy E at which Go' is evaluated, we obtain the solution of the Schrodinger equation in I which matches onto the solution in 11.

To derive an effective Schrodinger equation from (16) we vary the trial function q to minimise E. As E is then stationary with respect to small changes 6q, q must satisfy:

= E&) r in I , (17)

where n is the component of rperpendicular to S. We see that G,'(rs, rk) is an effective surface potential operating on q, and ( E -&)a G&'/de is a correction to give the embed- ding potential at the actual energy eigenvalue E : for complete accuracy we should re- evaluate Go' at E. It is interesting that aGo'/a&, which we obtained initially from the

A method of embedding 3799

volume integrals through I1 in (4), provides this energy correction to the embedding potential-this is very reminiscent of pseudopotential theory (Heine 1974). The normal derivative term 16(n - ns) dldns ensures that the effective Hamiltonian is Hermitian when it operates only within region I. The eigenfunctions and eigenvalues of this equation, defined entirely within region I, are solutions of the full Schrodinger equation for I embedded in 11.

3. Bound states

To illustrate the variational principle (16), we shall use it to find the bound states of a square well, radius rs, depth V. The square well itself constitutes region I, and the constant potential outside, region I1 (figure 2). The Green function in I1 which has zero

I I I

I I I " I

- r, - Figure 2. Square well

derivative on the surface of the square well has an s-wave component given by (Inglesfield 1971):

where $ = - 2 ~ , the energy measured from the constant potential, so the effective potential for embedding I in I1 is:

dGC'(rs, rs) - 1 1 - a& 4n 2rf y '

For the bound state we shall take a trial wavefunction in I of the form

q(r) = (sin k,y)/r. (21) Then with rs = 2 au, V = 1 au, and taking an arbitrary value of ko = 0.8, we obtain the bound state energy shown in table 1 for different values of E . The actual 1s bound state energy is E = -0.377 au, and we see that the values given by (16) are a fair variational estimate of this.

To proceed further we expand the wavefunction in I in terms of a set of functions XI :

~ ( r ) = 2 i aixi(r>. (22)

3800 J E Inglesfield

Table 1. Variational principle applied to square well.

-0.6 -0.4 -0.2

-0.292 -0.305 -0.302

The trial wavefunction in I is (sin kor)lr with k,, = 0.8 au. E is the energy at which the embedding potential is evaluated. V = 1 au. rs = 2 au.

Then substituting (22) into (16) and minimising E with respect to variations in the a,, we obtain the matrix representation of the embedded Schrodinger equation (17):

E [ H , + ( E - ~ ) F , l a , = E Op, , (23) I 1

where the Hamiltonian matrix is given by:

the energy correction to the effective potential by:

and 0, is the overlap matrix:

W e want to choose a set ofx, with sufficient flexibility to give the required values of q(rs) and dq(rs)lans-this means that we should not choose a set of states satisfying particular boundary conditions on surface S.

In the case of our square well let us take:

x , ( r ) = (sin k,r ) / r with k , = n d r * (27)

-07

~

- 0 5 0 - 0 4 5 -0 40 -0 35 -0 30 -0.25 -020

E b u l

Figure 3. Energy of 1s state. E,, , as a function of E . the energy at which the embedding potential is evaluated. V = 1 au. rx = 2 au.

A method of embedding 3801

The matrix elements then become:

HI, = (Ikf - V ) I

sin(k, - k,)rs - sin(k, + k,)rs) k , + k,

+ Isin k,rs [k, cos klrs - (sin k,rs)irs]

+ sin k,rs sin k,rs (1 + y ~ ) I 2 r ~ .

Fll = - (sin k,rS sin k,rs)/2y

1 ( i = j )

(i 4, ) J sin(k, - k,)rs - sin(k, + kl)rs) ' i k, - k, k, + k,

Taking four basis functions, with ro = n in (27), our matrix equation gives the Is bound state energy shown in figure 3 as a function of E , the energy at which G i l is evaluated. We see that the minimum energy is E = -0.377, in agreement with the exact bound state energy.

4. The Green function and continuum states

Jn the continuum range of energies, the embedding potential is complex and the Green function is the most useful quantity to calculate. The Green function of the embedded Schrodinger equation for I satisfies:

8C(rs, r ' ; E ) - 1 VfG(r, r ' ; E ) + t 6(n - ns) + V(r)G(r , r ' ; E )

ans

+ 6(n - ns) Is d*I'+S{'(rs, ti; E ) G(ji, r' ; E ) - EG(r, r ' ; E )

= 6(r - r ' ) r , r' in I (29)

-we evaluate the embedding potential Gi l at E so that the term in (17) involving aGi ' /a& disappears. This Green function is in fact identical to the Green function for I joined on to 11.

To show this we shall work with a finite region I1 so that all states are discrete. In terms of the exact eigenfunctions and eigenvalues of the system I t I1 the Green function is given by:

where the Y, are normalised to unity over I + 11. The Green function satisfying (29) can itself be expressed in terms of the complete set of states which are eigenfunctions of the corresponding homogeneous equation, with the energy at which Gi ' is calculated kept

3802 J E Inglesfeld

-the q1 in (32) are normalised to unity over I. It is clear from (32) that the poles of G occur when:

E = &(E) , (33)

in other words at the eigenvalues of a (- d v; + d 6(n - ns) a,) + V(r)q,(r)

+ S(n - ns) d2r&Gi’(rs, 4 ; E,) q,M) = E,ql(r). (34)

But this is the exact embedded Schrodinger equation (cf (17)), SOE, =Ef t” , and the poles of G are the same as the poles of GI+ 11. Moreover the residue of G at E, is:

(35) %Yr)qdr’) = q: (r)ql(r’) 1 - aE,/aE

from (15) the denominator normalises the wavefunction to unity over I + 11, so:

and the residues of G are just the same as the residues of GI 11. We conclude that G is identical to GItI17 when r and r’ are in I , and (29) gives us the Green function for I embedded in 11.

As before, we can solve (29) by expanding G in terms of some set of functions:

G(r, r’> = 2 G,lx,(r)x,(r‘) r, r’ in I. (37) 11

Substituting this into (29) gives:

7 G,( - 1 Vtx(r) + 4 6(n - ns) - a ~ l ( r s ) + v(r)x,(r) ans

+ W - ns) I, d2Wi’ ( r s , 4; E)xX4) - Ex,{r)) x,(r’)

= S(r - r ’ ) , (38) and multiplying this equation by Xk(r)x,(r‘) and integrating over rand r’ through region I we obtain:

2 Gi,(Hki - EOki)Oll = Ok17 (39) 11

A method of embedding 3803

where the Hamiltonian and overlap matrix elements are given by (24) and (26). From (39) we see that:

E (Htk - EOk)Gk, = 6,1,

G, = ( H - EO),'.

(40) k

so the matrix representation of the embedded Green function is:

(41)

In the continuum range of energy it is useful to work with the local density of states a(r, E ) (Heine 1980), that is the charge density of electrons with energy E-this can be found directly from the Green function:

4 r , €1 = E Iv,(r)IZ 6 ( ~ - E,)

= (1/n) Im G ( r , r; E + iE). (42)

Hence

with the embedding potential Gi ' in the Hamiltonian evaluated at ( E + ie). As an example of this formalism we shall again consider the square well. To find the

s-wave local density of states inside the square well we use the matrix elements given by (28); above the zero of energy we take Gi l t0 be:

Go'(rs, rs; E ) = (1/4n) (1 - iws)/2r$ K = (2E) *I2 . (44)

The results which we obtain for the local density of states integrated through the square well are shown in figure 4 over a very wide energy range, for different numbers of basis functions. We see that this method of finding the local density of states converges very well, even at high energies.

The Green function G, and embedding potential Go', have a branch cut along the

0.04 1

0 2 L 6 8 10 E lau)

Figure 4. Local density of states integrated through square well, n , as a function of energy E. V = 1 au, TS = 2 au. Full curve: four basis functions; broken curve: eight basis functions.

3804 J E lnglesfield

0 3-

02.

I s

C I

0 1

00 1

2 L 6 8 10 E lau)

Figure 5. Contribution to local density of states integrated through square well from poles in the analytic continuation of the Green function.

real energy axis in the continuum range of energy, but a resonance peak in Im G(E + iE) can correspond to a pole in the analytic continuation of G onto the unphysical sheet in the lower half E-plane (Csanak et a1 1971). This pole occurs at an eigenvalue E, of (34) in the lower half plane, with Go' analytically continued so that it is evaluated on the unphysical sheet at E,. In the case of our square well problem Go' is given by (44), and on the unphysical sheet in the lower half plane we take the sign of K so that it has a negative imaginary part. Using a matrix representation of the eigenvalue equation (34)-that is, (23) without matrix Fcorresponding toaG;'/d&-we then find solutions at

E, = (1.3, - 1.2) au E, = (6.2, - 3 . 2 ) au.

Each of these poles in the analytic continuation of G should give a Lorentzian contri- bution to the density of states in the square well, if we neglect the renormalisation term

Vacuum

S

Figure 6. Application of embedding method to surfaces.

A method of embedding 3805

(1 - aEjaE)-' in (3 .3 , of the form:

1 Im(Ei) nwell = - n[Re(E,) - E]* + [Im(EJ]" (45)

These are shown in figure 5, and comparing with figure 4 we see that they correspond remarkably well to the structure in the actual density of states, even for these s-states. This 'quasiparticle' approach should be particularly useful for dealing with virtual bound states in transition metal impurity problems (Zeller and Dederichs 1979).

5. Future applications

This method can be used quite straightforwardly to embed surface layers onto a substrate (figure 6). In this case, region I , where we solve the Schrodinger equation, is the top one or two layers of atoms and the vacuum; we simply add the surface potentialG;' on S to take account of the substrate. Go' can be found from the Green function for the infinite substrate crystal.

( a I ( b l

Figure 7. Application of embedding method to clusters of atoms. The embedding potential on surface S ( a ) can be transferred to S' (b ) .

To embed a cluster of atoms at first sight appears to involve a complicated shape for region I (figure 7a). However we can transfer Go' from S to give an embedding potential on the surface of a sphere, S' , (figure 76) with a flat potential in between. We then have the more straightforward problem of solving the Schrodinger equation for the cluster of atoms contained in a sphere. We have used this procedure to embed a cluster of Cu atoms into bulk Cu, and shall present our results in a later paper (Inglesfield 1981).

Acknowledgments

I have had useful discussions with Professor T B Grimley and Dr J B Pendry

References

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J E Inglesfield

Csanak Gy, Taylor H S and Yaris R 1971 A d o . A t . Mol. Phys. 7 287-361 Ellis D E, Benesh G A and Byrom E 1979 Phys. Rev. B20 1198-207 Garcia-Moliner F and Rubio J 1969J. Phys. C: Solid State Phys. 2 1789-96 Grimley T B and Pisani C 1974 J . Phys. C: Solid State Phys. 7 2831-48 Gunnarsson 0 and Hjelmberg H 1975 Phys. Scripta 1 1 97-103 Heine V 1974 Solid State Physics 24 1-36 (New York: Academic Press) Heine V 1980 Solid State Physics 35 1-127 (New York: Academic Press) Inglesfield J E 1971 J . Phys. C: Solid Stute Phys. 4 L14-7 - 1972 J . Phys. F: Met. Phys. 2 878-92 - 1978 Surf. Sci. 76 35578 - 1981 to be published Lang N D and Williams A R 1978 Phys. Rev . B18 61636 Pisani C 1978 Phys. Rev. 17 3143-53 Posternak M, Krakauer H, Freeman A J and Koelling D D 1980 Phys. Rev . B215601-12 Zeller R and Dederichs P H 1979 Phys. Rev . Lett. 42 1713-6