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Studies in perturbation theory
SPECTROSCOPY
Studies Part VII.
14, 119-136
(1964)
in Perturbation Localized
Theory
Perturbation*
Q,tantum Theory Project, L:nizlersity of Florida, Gainesville, Florida untl Quantum Chemistry Group, L;niz’ersity qf I:ppsala, Irppsala, Sweden
A perturbation is said to be localized if only a few elements of t#he energy matris are influenced in a specific representation. The corresponding Schrodinger ecluation can be transformed to a form which focuses the interest on those energy values as are changed by the perturbation. The associat,ed secular equation contains the energy as a nonlinear parameter, and it may be solved numerically by the second-order iteration procedure developed in a previous paper. The cases of a perturbation of a single diagonal element and a pair of nnndiag~~nal elements are treated separately as illustrative examples. The methcId is applicable to hetero-atoms in molecules, to impurities in solids and to molecular structures with “extra” bonds, etc. Numerical applications arc’ carried out elsewhere. I. ISTRODUCTION 12 prarturhation will be said to be localized if, in any particular choice of basis, only a few elements of the energy matrix will be influenced. Examples of localized perturbations are common and the simplest examples are the impurities in solids, heteroatoms in molecules, extra bonds in molecules, etc. The Schrodinger eyuation for a system with a localized perturbation can, of course, be solved by the standard perturbation theory. In this paper we will show, however, that there exists a much more convenient technique for treating this problem which will be called “localized perturbation theory.” The existence of such a method for treating inipurities in solids has previously been noticed by T
* Sponsored Knut tory,
in part by the King Gustaf T’I Adolf’s 70-Years Fund for Swedish Culture and Alice Wallenberg’s Foundation and in part by the Aeronautical Research Labora OBR through the European Of&e, Aerospace Researrh l’nited States Air Force. 11s
120 II.
LOWDIN
SOME SPECIAL
THEOREMS ABOUT OF THE ENERGY
LOCALIZED MATRIX
PERTURBATIONS
A. PERTURBATIONOF A SINGLE DIAGONAL ELEMENT Let US first consider the simplest type of a localized perturbation which can exist, namely, a change of a single diagonal element, say HI1 . The Schrodinger equation for the system has the form H9 = E* and, in order to go over to a discrete representation, we will introduce a complete orthonormal basis (&j and express the wave function in the form q = &&Ck . In this representation the Schrodinger equation can now be given the form HC = EC.
(1)
The problem is now to find out how a change in the single diagonal element H,, will influence the eigenvalues E. According to the partitioning technique developed in the previous paper, see, e.g., (17) in PT I, one has the relation
Hbb)-lHbl ,
E = HU + Hlb(E.lbb -
(2)
where (b) is the subspace which is the orthogonal complement to the function @I . In the following, it is convenient to normalize the wave function so that (\k 13?) = CtC = 1. Carrying out a partial differentiation of formula (2) with respect to E, one obtains aE/dHll = 1 -
&(
E . &,b -
&~)-2&l~E/dH11 .
(3)
According to Eq. (18) in PT I, one has further ‘!b = (E. &,b - &,-l&$l , which giVeS dE/‘aHn = [l + &,(E.&,b
-
?&,&2&,l]-1 = [l + ctbCb/j C, 12]-’ = /‘?I I2 (4)
Hence one obtains dE/dHll = /Cl 1’
(5)
indicating that 0 5 aE/dHII 5 1. This theorem implies that, if a single diagonal element HII is changed, all energy values E are changed in the same direction, unless they are entirely unperturbed. Since the trace of the energy matrix is invariant, the sum of the energy changes must equal the total change in the element Hn . The energy change in a particular eigenvalue E is, in a first approximation, proportional to the absolute square of the coefficient of the basic element +I in the eigenfunction P associated with the eigenvalue E under consideration. It is interesting to observe the behavior of the eigenvalue when the diagonal element HII becomes infinite ( f 00), whereas the remaining elements of the energy matrix stay finite. According to Eq. (5) in PT VI, contraction of the original secular equation with respect to the subspace (b) gives Hbb
=
Hbb
+
Hb&b/(E
-
HII).
(6)
STUDIES
IX
PERTURBATIOS
THEORY.
VII
11’1
Hence, one has the result
for all finite values E. This gives the theorem that, if a single diagonal element, becomes infinite, the eigenvalues of the entire matrix approach the eigenvalues of the truncated matrix obtained by striking away the row and column associated with the diagonal element involved. The theorem is valid even if the element of H1b would also become infinite, provided that the degree of infinity is of a lower order so that the quantity HlbHhl/H1l goes to zero. By contracting
the original secular equation to the subspace spanned by the
single element a1 , one gets finally E = HI1 showing that an eigenvalue will bccome infinite with H,, during the limiting procedure. It is instructive
to study the behavior of the eigenvalucs as functions of II,,
as shown in Fig. 1. We note that all the cigenvaluw arc monotonously increasing functions of the parameter H 11, and that there is a series of horizontal asymptotes corresponding to the eigenvalues of H bb. There is further the asymptote E’ = HI1 . We note that, according to the theorem of tract invariance the sum of the deviations of the eigenvalues from these asymptotes equals to zero. Formula
(5) has here been derived by means of the partitioning
technique
starting out from Eq. (2‘). We will here give an alternative derivation starting out from El. (1) which is useful also in other connections. The eigenvalue E
LOWDIN
122 may be expressed in the form
E = C+HC,
(8)
where we have used the condition C+C = 1. Taking the derivative of this relation with respect to an arbitrary parameter a, one obtains aE _ = C+,“%C+zHC+C+Hg acY (9) = C+ 2
C + E 2
(C+C)
= C+ ;G C.
This relation is often called the generalized Hellman-Feynman theorem. [For references, see Lowdin (3) .] Putting 01 = Hkk , one obtains particularly aE/aHkk
= ct(aH/aH,,)c
= C*BCk= 1Ck 12,
(10)
which is the theorem desired.
B. PERTURBATION
OF A KONDIAGONAL
ELEMENT
The energy matrix H is Hermitian which implies the property Hlk = Hzl . We will now study the change of the eigenvalues in the case when there is a perturbation of a single nondiagonal element, i.e., a perturbation of a pair of nondiagonal elements HM and H okwhich are situated symmetrically with respect to the diagonal For the sake of simplicity, we will assume that the perturbation is real so that one has the symmetry relation 6Hlk = BHkl . Putting o( = Hkl into the general equation (9)) we get aE/aHaI
= c+(aH/aHbl)C
= c*kcI
+ c*tzI, )
(11)
which is the basic theorem desired. It tells us that the change in a specific eigenvalue under variation of an element Hk2 depends on the product of the coeficients Ck and Ct by means of which the basic elements cPkand 91 enter the wave function 9 associated with the eigenvalue E under consideration. Because of the theorem of trace invariance, some of the eigenvalues are increased and other decreased under the variation of nondiagonal elements, since the sum of the changes in the eigenvalues have to be equal to zero. This theorem is also easily proven by summing the relation (11) over all the eigenvalues of the matrix and by using the orthonormality relation CC+ = 1. Formulas (10) and (11) are directly applicable to nondegenerate eigenvalues, and the treatment of a degenerate eigenvalue requires a few more words. The reason is, of course, that, for a degenerate level, there is an ambiguity in the choice of the eigenfunctions and hence in the associated coefficients CI;. The degeneracy may further be split by a perturbation and, in this case, it is convenient to consider the sum of the energy levels which arise from the same
STUDIES
IX PERTURBATION
THEORY.
VII
123
degenerate level. It is then easy to prove that the derivative of this sum with respect to a diagonal element Hkkequals the sum of the absolute squares 1C’k )‘, which quantity is independent of the choice of eigenfunctions in describing the degeneracy. Similarly, the sum of the eigenvalues associated with the degeneracy can he differentiated with respect’ to H~Iand the result equals the sum of the quantities ( (‘*/;(II + C,*C,) over the degeneracy, and again this quantit’y is independent of the choice of eigenfunctions. III. ht
us now
in the form of t#hcform
LOCALIZED
PERTURBATION
THEORY
study a perturbation problem where the Hamiltonian
is given
H = Ho + T'. Instead of (I), we now obtain a Schrbdinger equation il2)
(Ho + V)C = EC,
where V is the perturbation matrix. We will assume that this matrix is localized in the basic system j&j, and that only a finite number of matrix elements are essentially different from zero. One can rewrite ( 13) in the form (E. 1 - Ho)C = VC and, if the eigenvalue E under consideration is not simultaneously an eigenvalue to H,,,the operator (E.1 - Ho) has an inverse. Hence, one can derive t>herelation f 13) C = (X.1 - Ho)-‘VC. In the following it is convenient to introduce the notations K = (E.1 -
Ho),
G = (A.1 -
Ho)-’
Instead of the original equation (la), equations of the form (1
-
( l-1’) = K-l.
(l.ii
we obtain in this way a linear system of GV)C = 0.
(16)
This homogenous system has a solution only if the associated determinant is vanishing, which leads to the secular equation det (1 -
GVJ = 0.
(Ii)
It is now easy to show that, if t8henonvanishing elements of the perturbation V form a finite matrix of order p, the secular equation (17) is reduced so that it contains a determinant of only this order. We observe that such eigenvalues as are not influenced by the perturbation V will not occur as roots to the eigenvaluc problem i 17 1. This follows from the relation dct 11 - GV} = det { 1 - KP’V} = tlet [KP1(K -
V)) = det K-‘.dct det {l -
(K -
V), i.e.,
GV) = tlct (13.1 -
H}/det
(E.1
-
Ho),
(18)
LOWDIN
124
which shows that the eigenvalues which are common to H and Ho will not occur in the transformed secular equation (17). If the original secular equation for Ho is of finite order, one can hence be sure that the transformed equation (17) should be of the same or lower order. In an algebraic treatment, one has to be careful to utilize the cancellation of common factors in the numerator and denominator of the right-hand side of Eq. (18), since otherwise the order of the algebraic equation will blow up. Examples of this type were found in the numerical applications reported in a following paper (4). Evaluation of the inverse matrix G = (E- 1 - Ho)-‘. Some of the methods available for forming the inverse of an operator have been listed in PT IV, and here we will only briefly review the situation. Of primary importance in the practical applications is, of course, the method based on the solution of an equation system GX = 1 and the numerical and analytical tools for treating such systems. One should further observe the existence of the determinant formula Gkl = 11K l/Ik/llK 11, where )I K jj~k denotes the cofactor of the element Klk in the determinant )I K 11.Other methods for evaluating the inverse are based on the fundamental identity (A -
B)-’
= A-’ + (A -
B)-‘BA-l.
(19)
Repeated use of this formula leads to the infinite expansion
(A - B)-’
= A-lLg
(BA-‘)“,
(20)
which is convergent, if and only if all the eigenvalues of the matrix BA-’ are absolutely less than 1. The power series method has hence a very limited range of applicability. Since the matrix K = E. 1 - Ho can be split into two matrices A and B in many different ways, there are also many modifications of the power series method. If the difference (A - B) is a Hermitian or normal operator, there exists a unitary matrix U which brings it to diagonal form 3. so that Ut(A - B)U = 31. This leads immediately to the relation (A -
B)-’
= UZ’U~,
which is often used in localized perturbation one obtains (A -
B)$
theory.
= c UJJta& a
Other useful methods listed in PT IV the infinite product, and the successive In connection with the treatment of the Chebyshev expansions are further nique is based on the two formulas
(21) For a specific element, -
(22)
are the second-order iteration procedure, inversion technique. localized perturbation of cyclic systems, of fundamental importance. The tech-
STUDIES
IX PERTURBATION
(1 - BA-‘)-1
= &
C
THEORY.
1 + 2
l“5 _I
VII
1
Y’CI:(L~BA-~)
k=1
(23)
Here LY> 2 and 1’ = >$[a - (CX’- 4)1’2] are parameters which are chosen so that the expansions become convergent. The Chebyshev polynomials of the variable 2 = 2 cos 0 are defined by the standard relations C,(z) = 2 cos n. 8, S,,(s) = sin (,a + l)O/sin 0. The chebyshev expansions are partiuclarly convenient in treating matrices Ho having cyclic character, since they lead to nice closed expansions which are then valid also outside the range of convergence of t,he original expansions (5). It should finally be mentioned that in a continuous representation, there are further methods available based on the use of the resolvent and the associated kernel. The situation becomes particularly simple when Ho is associated wit,h free particles. In the one-dimensional case, there is also a simple treatment based on the use of the Wronskian in solving an inhomogeneous system which leads to an explicit formula for the operator (E - H,J)-l. IF‘.
SUMERICAL
TREATMERT
OF
SOME
SIMPLE
CASES
h order to illustrate the method, we will give a couple of simple examples. WC will start, with the case that a single diagonal element has obtained a finite perturbation so that the perturbation matrix has the form
(35)
In this case, there is a single nonvanishing element and one has p = 1. Equation (16) may be written in the form C = GVC or C’k = ~~~~Gfi,l~~,pCp and, in this case, it simplifies to the form C’s1 For 1~= 1 one obtains particularly
which is the relation found by Koster and Slater (1) and by Lax (2). If the inverse operator G is evaluated by the eigenvalue method and U is the unitary matrix which brings (E. 1 - HO) to diagonal form, one has according to (22) (28)
126
Substitution into (27) leads to the equation
ca I Ul, 12/(E-
E&O’) =
l/VII )
(29)
which relation has been carefully studied by Koster and Slater for the case of a linear chain. We note that for the case of a finite or infinite cyclic chain, the method using Chebyshev polynomials is a convenient method for obtaining a closed formula for the matrix G(E) and particularly the element Gn . Apparently it is possible to solve Eq. (27) by studying the function y = Gn(E) and determining the points for which it equaIs the value y = l/VI, . This can conveniently be done graphically, see Fig. 2. We note particularly that since G(E) = (E.1 - Ho)-‘, one has G’(E) = -(E.l - HO)-‘, which implies that the derivative is a negative definite matrix and that Gil(E) < 0. We note the close analogy between Fig. 1 and Fig. 2. Single nondiagonal matrix element perturbed. Let us now consider the case of a perturbation of a single nondiagonal matrix element, say HIr . The perturbation has the form 0 VI.2 . . . v = i
v21
0
.-.
.
.
. . .1
)
1721= I&*,
(30)
and one has p = 2. The secular equation (17) takes the explicit forIn
1
1 -
Gla VZl ;
-
Gn 1’11
-
Gzz Vx ;
1 -
c:,, I’IZ
= ”
(31)
from which follows the relation I l/V,, -
GUII = dG1:G22 ,
(32)
STUDIES
IN PERTURBATION
THKORY.
127
1.11
--’ FIG. 3
which lends itself to graphical treatment and to accurate solution by using tables of the matrix elements of G and interpolation formulas. It is evident that these simple examples occur only in problems of a rather restricted type. The case of a perturbation of a single diagonal element may occur, for instance, in a simplified molecular-orbital treatment of a molecule or crystal having a single heteratom or impurity, which is then causing a perturbation of the Coulomb integral for the associated effective Hamiltonian. The case of a perturbation of a single pair of nondiagonal elements may occur in a simplified molecular orbital treatment of a ring molecule with an extra bond, e.g., naphthalene considered as a IO-membered ring perturbed by an extra bond. Some of the simplest systems which can be treated by this approach are examplilied by the molecules in Fig. 3. Y. NUMERICAL
TREATMENT
OF TRANSFORMED IN GENERAL
SECULAR
EQUATIOX
In the case of a more general localized perturbation, the numerical treatment is more complicated. For the solution of the transformed secular equation (16)) it>is often convenient to use the methods developed in a previous paper, PT VI. In localized perturbation theory, the basic equation is of the type MC = 0,
(33,
where M = 1 - GV = 1 - (A’. 1 - Ho)-lV. Th is means that the matrix rlement of M depends on the eigenvalue parameter fi in a rather complicated way. Before we start, we note that the coefficient matrix (1 - GV) is usually not self-adjoint, which implies a difference against the case treated in PT VI. Since V is a localized perturbation, the corresponding matrix V may after a suitable reordering of the basis be written in the form
v=
v
( > AA
0
0
0’
(34)
where V..,.., is the smallest quadratic matrix containing all the actually nonvanish-
138
LiSWDIN
ing elements of V. Let the order of this matrix be p. Making a corresponding partitioning of the matrix G and the vector C: C
=
C.4
0
(35)
I
CL3
one obtains from (33) : (L
-
G,,VdC,
= 0;
(36)
Cg = -GsaV_uCa
;
(37)
which relations are analogous to (26) and (27). The first equation is again of the form (33)) but it is now restricted entirely to the subspace (A), whereas the second gives the subveetor CB in terms of Ca . We can now proceed by considering the subspace (A) alone. Since the submatrix (lAA - GAAVAA)is not self-adjoint, it is convenient to multiply relation (36) to the left by Vaa , so that one obtains (V,, - VAaGaaVAa)CA = 0. Introducing the notation N AA = Vu
-
Va,G_.iaV,,
,
(38)
N AA = VA* -
VaaGuVaa
,
(39)
one has
where NtAI = NAa . Since det {NAAJ = det (V,,] det { IdA - GaAVAA), the associated secular equation will vanish identically for all values of E in the special case det {V,,) = 0. In this case, the equation system (39) is under-determined, and the approach has to be modified. In the following, we will assume that det IV,,) # 0. For the dependence of NAl = NaA (E) on the eigenvalue parameter E, one has N ** = VUN&a(E)
VaaGaa(E)Va.~
= -Va..&(WL
,
(40)
= VaaG?i,V,a
= (G,uV.a)t(G.uVau).
The last relation shows that the derivative NLa(E) is a positive definite matrix. Application of the general formula (21) in PT VI leads to a second-order iteration procedure connected with the Newton-Raphson Method: E” = Eo -
(VA.4 -
V,,G,,V,,>,/(V,,G”,,vaa)o
= Eo -
CtA(VAA -
V,,G,,V,,)C,/Ct,V,,G~~V~~C~.
(41)
Several modifications of this formula are useful. It is often convenient to introduce the vectors fA = VaaC, and d, = GA.4f_4 = GaaVA,CA, which gives E* = -7%- fta(Ca
-
da)/dt,da ,
(42)
HTCDIER
I?i PERTUKBATIOK
THEORl-.
PI1
129
which is a convenient form for iteration, since it involves only vectors of the same order as the localized perturbation V. In connection with the numerical evaluation of this and similar expressions, we note the economy of using vectors instead of matrices, whenever possible. The iteration procedure has been described in 1’1’ VI, and we will hew only briefly review the method. One starts by dividing t,he subspace (A) characteristic for the localized perturbation V into two arbitrary subspaces: one (a j con taining a single basic element, say & , and the remainder (b). This leads t’o the following partitionings :
The iteration procedure involves the following steps: 1. I%timat,e a starting value E = E. . 2. (lalculate the matrix Na, = N,,(E) for E = E!‘o, where NAa = VA, G.,.,(K)V,, . We note that this matrix is of the same order p as the localized pt~rturhat~ion. The matrix G(E) refers only to the unperturbed system characterizcd by the matrix H, , and this inverse matrix may hence be evaluated otw for all as a function of the parameter E. The situation becomes particularly simple for cyclic systems since there exists a closed form for G in terms of Clhebpshw polynoiuials. 3. (‘alculate the subvector CL” of C, by solving the equation system N&b = -NblCl Thc~ value of C1 is arbitrary and one may choose C1 = 1. c, and evaluate the vector d,, = G.4.4VA.IC.L. 1. (‘onstrlict the vector C., = C/l 0 5. (‘alculate E* according to formula (-42) or the alternative formula E* = E’u -
C~*[N,,C,l,/dt.~d.~
.
(II
I
A. Repeat the procedure, starting from E = E*. It should be observed that the iteration procedure renders both the eigenvalues and the cigenfunctions. However, only those eigenvalucs which are perturbed by the localized perturbation V will be obtained in this way. Once the subvect’ol C., has been evaluated, the subector CH is obtained by using the relation (.?i,.
By means of the localized perturbation method, one can carry out a transformation of the secular equation for a perturbed system to such a form that one focuses the interest on those energy cigenvalues as are influenced by the perturbation. X perturbation is said to be localized if only a finite number of its matrix elements is nonvanishing, so that one can take out a finite matrix of order p which contains all the essential elements of the perturbation. The trans-
LiiWDIN
130
formed secular equation has the same order, and it can be solved by a secondorder iteration procedure which quickly gives the eigenvalues and the eigenfunctions. The method is particularly valuable in those cases where the resolvent (E. 1 - Ho)-’ associated with the unperturbed Hamiltonian Ho can be simply evaluated. In case Ho has a one-dimensional cyclic character, the resolvent may be expressed in closed form by means of Chebyshev polynomials. The calculation of the resolvent in the three-dimensional case is more cumbersome but has been treated by Calais and Appel (6). The powerfulness of the resolvent technique in the one-dimensional ease has also been noted by Gilbert (7). The localized perturbation method has been applied to a study of a series of heteroatomic conjugated systems by Ali and Wood (4). Further applications are in progress. Received
December
7, 1963 REFERENCES
1. G. F. KOSTER AND J. C. SLATER, Phys. Rev. 94, 1392 (1954); 96, 1167 (1954). 8. M. LAX. Phys. Rev. 94, 1391 (1954). 3. P. 0. LB~DIN, J. Mol. &e&y. 3, 46 (1959). 4. M. A. ALI AND R. F. WOOD, Technical Note No. 75, Uppsala Quantum Chemistry Group, April 1, 1962. 6. P. 0. L~WDIN, R. PAUNCZ, AND J. DE HEER, J. &zth. Phys. 1,461 (1960). 6. J. L. CALAIS AND K. APPEL, Technical Note No. 96, Uppsala Quantum Chemistry Group, May 1, 1963. (To be published, J. 2Math. Phys.) 7. T. L. GILBERT, “The Green’s Operator for the Finite Chain with Cyclic Boundary Conditions,” Research Note, Argonne National Laboratory.