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An alternative to orthogonalisation of wavefunctions and calculation of X-ray spectral intensities

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1977 J. Phys. B: At. Mol. Phys. 10 2531 (http://iopscience.iop.org/0022-3700/10/13/008) View the table of contents for this issue, or go to the journal homepage for more

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J. Phys. B: Atom. Molec. Phys., Vol. 10, No. 13. 1977. Printed in Great Britain. @ 1977

An alternative to orthogonalisation of wavefunctions and calculation of x-ray spectral intensities P L Khare, Pranawa Deshmukh and C Mande Department of Physics, Nagpur University, Nagpur-440010, India Received 25 October 1976, in final form 8 March 1977

Abstract. Simplifying assumptions made while considering complex systems often lead to non-orthogonal sets of wavefunctions. A procedure is suggested for employing such non-orthogonal wavefunctions without first orthogonalising them. The method involves the use of a basis reciprocal to the basis of the non-orthogonal functions. The relative intensities of Kcti, K/3, and KB2 x-ray spectral lines calculated by the proposed method are in fair agreement with available experimental data.

1. Introduction

Usually, one has to take recourse to several approximations while dealing with physical systems. In atomic physics, the central-field approximation has been extremely useful in obtaining valuable information about the atomic structure. The HartreeFock method of calculating self-consistent-field solutions for atomic orbitals (Slater 1960a) is undoubtedly an extremely powerful one in the central-field approximation. Although the numerical solutions of the Hartree-Fock integro-differential equations have their well known advantages and wide applicability (Slater 1960a, Rosen and Lindgren 1968, Meldner and Perez 1971, Hibbert 1975), their tabular character limits their utility in analytical calculations. Analytical solutions which can be expressed in the form of well known functions are more useful than the numerical ones from this point of view, though the various approximations involved in obtaining them often restrict their range of validity (Pratt et a1 1973). The concept of screening when used in conjunction with the central-field approximation is particularly useful in problems in atomic structure analysis and yields screened hydrogen-like analytical atomic orbitals. Such wavefunctions have been obtained using x-ray spectroscopic data and have been employed to calculate Auger energies, shake-off probabilities, energies of x-ray satellites etc (Tankhiwale and Mande 1970, Tankhiwale et al 1971, Tankhiwale 1971). However, on account of the different effective 2 values for the electrons in different orbitals, the one-electron wavefunctions are eigenfunctions of different Hamiltonians and are consequently nonorthogonal. While dealing with such functions the first step usually taken is to orthogonalise them (Morse et a1 1935, C J Roothaan 1960 see Slater 1960b, Tankhiwale and Mande 1970) using a suitable procedure. However, forming orthogonal functions which are linear combinations of eigenfunctions of different Hamiltonians is a mathematical simplification difficult to interpret physically. Moreover, forming these orthogonal combinations itself often makes the calculations cumbersome. 253 1

2532

P L Khare, P Deshmukh and C Maizde

In the present paper, a method is suggested to use the screened wavefunctions without orthogonalising them. In the proposed method, we make use of the set of functions reciprocal to the basis formed by the non-orthogonal wavefunctions. The use of reciprocal bases to deal with non-orthogonal wavefunctions was introduced by Des Cloizeaux (1960) who has discussed the transformation of an implicit eigenvalue equation into an ordinary eigenvalue problem by generalising the Lagrange formula to operators. Similar work has been carried out by Brandow (1967), Newman (1970), Moshinsky and Seligman (1971) and Kochanski and Gouyet (1975). A somewhat different and simpler approach is suggested in the present paper by making use of an approximation termed as ‘the non-orthogonal basis approximation’ in 42. To illustrate the method, we apply it to calculate the relative intensities of some x-ray emission lines and compare the results obtained with the available experimental and theoretical data.

2. The proposed alternative to orthogonalisation Let [xi] be a set of linearly independent non-orthogonal functions such as the screened hydrogen-like one-electron wavefunctions mentioned above. An arbitrary function $ in the concerned space of state functions can be expressed as a linear combination of these functions. When a normalised function $ is expressed as a linear combination of orthonormal functions ui as

$

=

1ciui i

then the physical meaning given by the postulates of quantum mechanics to the expansion coefficients ci is that lci1 represents the probability of finding the system in the eigenstate ui when the state of the system is represented by the function $. The state function $ can be expressed as a linear combination of the non-orthogonal functions xi as

’

The probability for the system to exist in the state xi cannot now be taken as iaii2, nor is Xi1ail’ equal to unity as is Ci1ci/’. In the case of the basis Cui], ci =

(ui,

$1.

However, in the case of the basis [xi], ai is not equal to reciprocal to [xi], given by the relation define a basis [4i] (Xi,

(3)

(xi, $). In

4j) = J i j

this case if we (4)

then the expansion coefficients in (2) are given by ai =

(4i5

$1

as can be seen by direct substitution. The set [4$] can also be used as a basis and $ can be expressed as

(5)

illternatice to orthogonalisation of warefunctioizs

2533

the expansion coefficients hi being given by bi

=

(Xi, $1.

(7)

As the function $ is taken to be normalised, we have

on account of the reciprocity of qhj and 1,. Equation (8) is the usual expression for the norm of a vector in tensor algebra. ($, $) can also be expressed as (Cjbjq5j,Ciaixi)giving

The two relations (8) and (9) are the analogues in the reciprocal bases [4+] of the relation 1

=

($, $)

=

[xi] and

1c:cj 1

for the orthonormal basis Cui]. When the base functions [xi] are assumed to represent the physical states of a system the following points are implied. (i) When a system is represented by one of the base functions xm, the probability of its existing in this state is unity and in any other base function is zero. (ii) When the system is represented as a h e a r combination of x k , there must exist a definite probability Pk for the system to exist in the base function l ( k , and C k P , must be equal to unity. Guided by the above considerations, we shall now express the probability PI in terms of the expansion coefficients ai and b,. The expression (aTbi) can be put in the following form, by using equation (7),

By analogy with the case of the orthonormal states, one would be led to take the probability of the system existing in the state xi (or in the associated state $i in the reciprocal basis) as arbi or (a,x,,$), if all the quantities (aiXi,$) were real and positive. However, when the functions xi are non-orthogonal, the values of (a&,$) may not always be real and positive. In such a case we make the approximation that the relative probability Pi for the system to exist in the state xi is given by

The above expression means that the relative probability of the system to exist in the states lI and q51 in their respective bases is taken to be the same, which is reasonable because x1 and qhl correspond to each other in the two reciprocal bases. We call (12) the ‘non-orthogonal basis approximation’. It is of interest to see under what condition equation (12) agrees with the postulates of quantum mechanics.

2534

P L Khare, P Deshmukh and C Mande

Now,

where C' denotes a summation over all j # i. In the case when the sum of the cross terms CJaTaj(xi,xj) is zero, then

and

If

[xi]

are normalised

which is in agreement with the postulates of quantum mechanics. While relation (12) may be regarded as a semi-empirical expression for Pi, and its validity will have to be tested in the light of experimental results, the following may be put forth in support of expression (12) for the probability. (i) If I) is a mixed state, the expression gives a definite probability as a real positive number for the system to be in each of the base functions xi. (ii) If the system exists in one of the base functions xm, the probability of the system to be in that state as given by (12) comes out to be unity and zero for being in any other state. (iii) As a special case, the expression (12) reduces for an orthonormal basis to the standard formula in quantum mechanics. (iv) The assumption that the non-orthogonal functions are good enough to represent the stationary states of a system is itself an approximation. When this approximation is used, a corresponding modification in the expression for the probability becomes necessary. (v) An appropriate test for the validity of the expression (12) used for the probability would be in agreement with experimental results. As will be seen in 93 this test is satisfied quite well for intensities of x-ray spectral lines. We now obtain an expression for the coefficients ai. Let the functions + j be expressed as a linear combination of the original base functions [xi] by the relation

The expansion coefficients Rki are the elements of the matrix of the operator R

Alternative to orthogonalisation of wavefunctions which transforms the basis

[xi] to the bases [ $ j ] .

2535

Hence

The matrix elements R k j can be determined as follows. Let S be the matrix generated by the overlap integrals (7. &I) 7.) = iJ

s...

(16)

IJ

Using equations (4), (14) and (16) we get /

\

Hence, R = S-'.

(18)

As the matrix elements (xi,x j ) of S are known, it is straightforward to calculate the coefficients R k j in (15) by obtaining the inverse matrix of S. Now, let a system which is initially in state xr be transformed by an operator I/ to a state $ given by $

= I/&.

(19)

The numerator in the expression (12) for the probability of a transition to a state x j will then be given by

By employing the appropriate operator to describe a given interaction, the probabilities of transitions induced by it can be calculated using the above relation. 3. Calculation of intensities of some x-ray spectral lines For the purpose of calculating intensities of x-ray spectral lines, we may restrict ourselves to non-relativistic considerations without any serious loss of accuracy (Heitler 1954). While calculating the line intensities, we shall be concerned with processes in which a single photon is transferred between a radiation field and an atom. The interaction Hamiltonian can then be written as

where e is the electronic charge, m the electronic mass, c the velocity of light. p the momentum and A the magnetic vector potential. In a spontaneous emission in

2536

P L Khure, P Deshmukh and C Mande

which the state of an atom changes from, say, the state a to the state b, emitting a photon of energy Aw, the probability amplitude PA is given by (Heitler 1954) ,PA = - i v / 2 n : v ~ ~ , b ( b / r r / u ) (22)

-

where the symbols have their usual meanings and the atomic units e = 1 = h have been used. In this expression is incorporated the dipole approximation, which even for x-rays of short wavelengths is expected to account for at least the dominant effects (Rooke 1974). We now use the above formalism to calculate the probabilities of the KM,,KP, and KP, transitions in yttrium ( Z = 39) and zirconium ( Z = 40). For these emission lines the final level of the electronic transition is the 1s state while the initial levels are 2p3,,, 3p3,, and 4p,,,, 3,2 (unresolved) respectively. To describe these states, we have obtained screened hydrogen-like wavefunctions by the method of Tankhiwale and Mande (1970). To take account of the different penetrations into the atomic core by the s and p electrons, we have introduced a slight modification over the method used by them. We have subtracted the factor rdZ,/dr (where r is the radius of maximum charge density of an orbital and Z , the effective nuclear charge for the energy of the corresponding state) separately from the Z , value for the s state and from the average 2, for the P , , ~and p3,, states, thus obtaining different values, Z,, for the effective nuclear charges for the s and p hydrogen-like orbitals. The Z , values (Damle 1966) and the Z, values obtained thus for yttrium and zirconium are listed in table 1. These Z , when substituted for Z in the hydrogen wavefunctions (Pauling and Wilson 1935) give analytical screened hydrogen-like wavefunctions. These functions then form a basis [xi] of non-orthogonal wavefunctions to which the treatment of $ 2 is applicable. In order to use these screened wavefunctions to calculate transition probabilities of the Kal, KP, and KP, spectral lines we first calculate the overlap integrals (xi,l j ) , where xi stands for a screened hydrogen-like wavefunction, thereby generating the matrix S of equation (16). Calculations of (xi,xj) are simplified by noting that one need calculate only those integrals for which the angular quantum number I is the same in xi and xj, as the angular parts of the wavefunctions in a central field are orthogonal. This facilitates the partitioning of the matrix S and obtaining its reciprocal R. Due to isotropic properties which characterise spontaneous emissions Table 1. The effective nuclear charges. Yttrium Level

Zirconium

Z,

Orbital

Zf

z,

Orbital

z,

35.13 26.28

1s 2s

38.71 35.65

36.05 27,13

1s 2s

39.68 36.38

2P

34.26

3s

26.41

25’91> 25.55 16.82

2p 3s

28.27

14.10

3P

24.50

3P

26.35

10.20 7.38

3d

22.70

24.73 16.14

5.66

4s

7.38

4P

5.66

:::;;>

34.98

21.47 7.65 5.66 5.66)

4p

5.66

2537

Alternative to orthogonalisation of wavefunctions

with which we are concerned here, it is sufficient to calculate the matrix elements - i, 271, w,b(bizlu), by considering the magnetic quantum number m = 0, for the purpose of obtaining the line intensities to be compared with each other. It is thus sufficient to consider only a restricted basis [xnro] of the hydrogen-like orbitals. The following basis (which though not complete appears to be adequate for the present calculations) has been used: Cxloo, ~ ~ ~ 2 ~ ~ x300, 0 0 , x310, , x 3 2 0 , x400, ~ 4 ~ 0 The 1 . matrix S for yttrium obtained by calculating the overlap integrals is given in table 2. The matrix for zirconium is similar. In order to calculate the coefficients a, which can be fed into the equation (12), the expressions of the standard perturbation theory (Powell and Crasemann 1961) are modified to take account of the non-orthogonality of the basis functions in the following manner. I (19)) can be expressed as The final state $(t) = H , n t ~(equation -

I

k

The standard expression for the coefficients ak(t) when using an orthonormal basis Cui] is

ak(t) =

~

Ek

Hkl

- E,

[ 1 - exp [ - i(Ek - E , ) t / h ] )

where Hkl = (uk, Hlntul).It can easily be seen that the time average a(k) of ak(t) over a period is simply Hkl/(Ek- El), In the present case, Hkl will have to be modified by taking Hkl= (&, Hlntxl).As a result of equation (14) the expression for H k l becomes

By feeding the values for R$ from the inverse of the matrix S, employing (22) to calculate the matrix elements (xf, HIntxr)and using equations (23) and (24) one gets the values for the coefficients ak(t). It is then possible to calculate the transition probabilities by making use in equation (12) of the coefficients ak calculated thus. It is to be remembered (Hedin 1974) that while calculating the matrix elements (xi,H,,,x,) in x-ray emission, there is a hole in the inner shell (K shell in the present case) as a result of which the efl'ective nuclear charge for the state x1 is enhanced. This enhancement can be estimatcd 'is has been done by Tankhiwale et al (1971) and comes out to be 1.02, 1.3 and 1.2

Table 2. The matrix S 1 0,043433 0 0,076452 0 0 0.04621 1 - 0

0.043433 1 0 0,22266 0 0 - 0'0347225 0

=

0 0 1 0 0,225323 0 0 0.061642

(xi, xJ) for yttrium 0,076452 0,22266 0 1 0 0 - 0.173 14 0

0 0 0,225323 0 1 0 0 -0'158061

0 0 0 0 0 1 0 0

0,046211 - -0'0347225 0 - -0.17314 0 0 1 0

-

0 0 0,061642 0 0.15806 1 0 0 1 4

2538

P L Khare, P Deshintikh and C Manrle

for the x ~ ~ 3~1 0and ~ x410 , orbitals respectively. The correction for the inner-shell vacancy has been taken into consideration in the present calculations. We have calculated the probabilities for 1 2 1 0 - + ~ i / 3 110o 0 - + ,~ l ~and ~ x410-);c100 transitions. It is important to note that the transition probabilities for the x210-+ xloo and x 3 1 o - + ~ processes ~ ~ ~ give only the total values for (Ka, and KaZ) and (KP, and KP3) transitions respectively, as no account of the spin-orbit splitting is incorporated in Coulombic screening. However, to calculate explicitly the intensity ratios Int(KP,)/Int(Ka,) and Int(Kj2)/1nt(Kxl), the results of the above calculations can be used as follows. It is known from the statistical weights of the energy levels under consideration that both Int(Ka2)/Int(Kxl)and Int(KP,)/Int(KP,) have a value 0.5 (Compton and Allison 1935). Consequently.

--

Int(3p -+ 1s unresolved transition) - Int(3p3,, -+ 1s) + Int(3pl12 Int(2p -+ 1s unresolved transition) - Int(2p3#, 1s) + Int(2p1,, --+

-

1s) 1s)

1.5 Int(KP1) 1 3 Int(Ka,)

- W K P 1) Int(Ka 1)

and Int(4p --+ 1s unresolved transition) - Int(KP2) Int(2p -+ Is unresolved transition) Int(Kal) Int(Ka