NON-STANDARD REPRESENTATIONS OF THE DIRAC EQUATION ...

NON-STANDARD REPRESENTATIONS OF THE DIRAC EQUATION AND THE VARIATIONAL METHOD

MONIKA STANKE AND JACEK KARWOWSKI

Instytut Fizyki, Uniwersytet Mikolaja Kopernika, Grudzi¸adzka 5, PL-87-100 Toru´ n, Poland

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example.

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MONIKA STANKE AND JACEK KARWOWSKI

1. Introduction The Dirac equation for an electron in the field of a stationary potential V reads h i c (α · p) + (β − I)µc2 + V Ψ = EΨ, (1)

where Ψ is the four-component Dirac spinor, α, β and I are 4 × 4 Dirac matrices, c is the velocity of light and E is the energy relative to µc 2 . Usually the variational principle is applied to Eq. (1) in the Dirac-Pauli representation 

V − E, c(σ · p) c(σ · p), V − E − 2µc2

ΨL ΨS



!

= 0,

(2)

where σ are 2 × 2 Pauli matrices, and Ψ L , ΨS are two-component spinors, respectively large and small components of the Dirac wavefunction. In the variational procedure ΨL and ΨS are expanded in predefined basis sets NL S NS {φL m }m=1 and {φm }m=1 , respectively. Thus, ΨL =

NL X

L L Cm φm ,

ΨS =

m=1

NS X

S S Cm φm ,

(3)

m=1

L }NL and {C S }NS are variational parameters. The basis funcwhere {Cm m m=1 m=1 q tions are usually taken in the form φ ∼ r g e−αr , where g and α may be treated as non-linear variational parameters, and q = 1, 2 depending on whether a Slater-type or a Gauss-type basis sets are used. The basis functions are usually centered on the nuclei and, in some cases, on other properly selected points, as e.g. on the geometric center of the molecule or the middle of a bond. The pattern of convergence of the variational energies Ej , j = 1, 2, . . . , N L , derived from the resulting algebraic Dirac equation

H LL − ES LL , H LS H SL , H SS − ES SS where

L LL Hmn = hφL m |V|φn i,

!

CL CS

!

= 0,

(4)

SS = hφSm |V − 2µc2 |φSn i, Hmn

LS S Hmn = c hφL m |(σ · p)|φn i,

aa Smn = hφam |φan i, a = L, S,

to the corresponding exact eigenvalues E j of the Dirac Hamiltonian has been studied by many authors (see e.g. [1–3] and references therein). It has been demonstrated that the relations between spaces H L and HS spanned NL S NS by the basis functions {φL m }m=1 and {φm }m=1 respectively, are crucial for the correct behavior of the variational solutions of Eq. (4). In particular,

REPRESENTATIONS OF DIRAC EQUATION

3

the solutions of Eq. (4) converge to their non-relativistic counterparts while c → ∞ if HS = (σ · p) HL . (5) Besides, condition (5) is necessary for E to be an upper bound to the corresponding eigenvalue E. The question how to control the behavior of the variational energy by using rather weakly constrained variational trial functions, motivated the formulation of a number of minimax principles [4–6]. A detailed discussion and classification of these approaches has been given in ref. [7]. In most general terms, they are based on the following condition: E = min {L}



max hΨ|H|Ψi , {S} hΨ|Ψi 

(6)

where E is the ground state of a Dirac electron and {L}, {S} refer, respectively, to the spaces in which large and small components of the Dirac-Pauli wavefunction are represented. It is easy to see that Eq. (1) is invariant with respect to a similarity transformation in the four-dimensional spinor space. If Ψ and E are solutions of Eq. (1) then h

i

˜ = E Ψ, ˜ ˜ · p) + (β˜ − I)µc2 + V Ψ c (α

(7)

˜ = AΨ, and A is a non-singular 4 × 4 ˜ = AαA−1 , β˜ = AβA−1 , Ψ where α matrix. However, conditions (5) and (6) are not invariant with respect to such a transformation. Therefore, the performance of a variational procedure applied to the Dirac equation depends on the selected representation. The aim of the present work is to study this dependence. Hartree atomic units are used in this paper, however in some places the mass µ of the electron is written explicitly in order to make the presentation more clear. 2. The Dirac-Pauli representation The Dirac-Pauli representation is most commonly used in all applications of the Dirac theory to studies on electronic structure of atoms and molecules. Apart of historical reasons, there are several features of this representation which make its choice quite natural. Probably the most important is a well defined symmetry of ΨL and ΨS in the case of spherically-symmetric potentials V. The Dirac Hamiltonian H = c (α · p) + (β − I)µc2 + V

(8)

does not commute with the orbital angular momentum operators L 2 and Lz , however it does commute with the total angular momentum J = L + 12 Σ,

(9)

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MONIKA STANKE AND JACEK KARWOWSKI

where, in the Dirac-Pauli representation, Σ=



σ, 0 0, σ



.

(10)

Also, both H and J commute with an operator K defined as K = β (Σ · L + I) ,

(11)

where I is a 2 × 2 unit matrix. Explicitly K is given by K=



k, 0 0, −k



,

(12)

where k = (σ · L) + I acts in the two-dimensional spinor space. Thus, a four-component eigenfunction of H may be chosen to be a simultaneous eigenfunction of J2 , Jz and K. The corresponding eigenvalues are j(j + 1), mj and k = ±(j + 1/2). Then, for a spherically-symmetric potential V, we can write Ψ = Ψnkmj , where n is the quantum number associated with the radial variable. Eq. (12) implies [(σ · L) + I] ΨL nkmj [(σ · L) + I] ΨSnkmj

= k ΨL nkmj , = −k ΨSnkmj

(13)

Since L2 = J2 − (σ · L) − 3/4, we can rewrite Eqs. (13) as L2 ΨL nkmj L2 ΨSnkmj

= k(k − 1)ΨL nkmj = k(k + 1)ΨSnkmj

= lL (lL + 1)ΨL nkmj , = lS (lS + 1)ΨSnkmj .

(14)

The orbital angular momentum quantum numbers, l L ≡ l and lS , corresponding, respectively, to the large and to the small components of the Dirac spinor are equal to l = lL = and S

l =





k − 1 = j − 1/2, −k = j + 1/2,

k = j + 1/2, −k − 1 = j − 1/2,

if if if if

k > 0, k < 0, k > 0, k < 0.

(15)

Hence, lS = l + 1 if k > 0 and l S = l − 1 if k < 0. Consequently, in L the Dirac-Pauli representation ΨL and ΨS have definite parity, (−1)l and S (−1)l respectively. It is customary in atomic physics to assign the orbital angular momentum label l to the state Ψ nkmj . Then, we have states 1s1/2 , 2s1/2 , 2p1/2 , 2p3/2 , . . ., if the large component orbital angular momentum quantum numbers are, respectively, 0, 0, 1, 1, . . . while the corresponding small components are eigenfunctions of L 2 to the eigenvalues 1, 1, 0, 2, . . ..

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REPRESENTATIONS OF DIRAC EQUATION

According to Eqs. (2) and (13) the Hamiltonian eigenfunction in the Dirac-Pauli representation may be written as 

Ψnkmj = 

ΨL nkmj ΨSnkmj



L Θkmj Rnk

1 = r

S iΘ−kmj Rnk

!

,

(16)

L , RS are radial where Θkmj , Θ−kmj are the spin-angular functions and R nk nk amplitudes. As one can easily check (see e.g [8]),

(σ · p)ΨL nkmj (σ · p)ΨSnkmj

k d L − Rnk dr r   k d 1 S − − Rnk = Θkmj r dr r = iΘ−kmj

1 r





(17)

Consequently, the angular dependence may be removed from Eq. (2) reducing the Dirac equation to V − E,

c (d/dr − k/r) ,

!

c (−d/dr − k/r) V − E − 2µc2

L Rnk

!

S Rnk

= 0.

(18)

Changing the representation, i.e. taking ˜ = Ψ

!

=

A=



˜u Ψ ˜ l, Ψ

where

aLL ΨL + aLS ΨS aSL ΨL + aSS ΨS aLL , aLS aSL , aSS



!

,

,

and aLL , aLS , aSL , aSS are 2 × 2 matrices, destroys the symmetry relations between the components of the Dirac spinor. If we restrict our considerations to the case of aLS aSL 6= 0, then the Dirac-Pauli representation is the only one in which, for spherically-symmetric potentials, the components of the Dirac spinor are eigenfunctions of L 2 . Another important feature of the Dirac-Pauli representation is its natural adaptation to the non-relativistic limit. If |V − E|