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Relativistic Density Functional Theory Method: Applications to Autoionization States of Atoms

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Relativistic Density Functional Theory Method: Applications to Autoionization States of Atoms Turk J Phys 30 (2006) , 231 – 243. c TUB¨ ITAK˙ Relativistic Density Functional Theory Method: Applications to Autoionization States of Atoms S¸akir ERKOC¸ Department of Physics, Middle East Technical University, 06531 Ankara-TURKEY e-mail: [email protected] Received 18.07.2006 Abstract A briefintroductory information about the accurate atomic calculations using Density Functional Theory has been reviewed. How relativistic effects can be considered in atomic calculations through the density functional theory has been discussed. Some sample calculation results for the autoionization states ofthe neutral and cationic atoms have been presented. 1. Introduction The Newtonian assumption that the probe (light or photon) does not interact with the object of mea- surement is not valid in the microscopic world, because of uncertainty principle. Quantum mechanics was invented to overcome the difficulties introduced by the uncertainty principle. This is what leads to a prob- abilistic picture in quantum mechanics. The fundamental equation in quantum mechanics, the Schr¨odinger equation, Hψ = Eψ, is obtained by inserting the operators for the position and energy in the classical →− ∂ → ∂ Hamiltonian energy expression; pα i¯h ∂α ,α= x, y, z; E i¯h ∂t. The Hamiltonian for a particle is given in classically, by 1 H = (p2 + p2 + p2)+V (1) 2m x y z substituting the appropriate operators for the momenta and energy we get the time–dependent Schr¨odinger equation: ¯h2 ∂ψ ∂ψ ∂ψ ∂ψ − + + + Vψ= i¯h (2) 2m ∂x2 ∂y2 ∂z2 ∂t If ψ does not depend on time, then the energy E is a constant of the system, then we obtain the time– independent Schr¨odinger equation: ¯h2 ∂ψ ∂ψ ∂ψ − + + + Vψ= Eψ (3) 2m ∂x2 ∂y2 ∂z2 All quantum mechanical expressions, operators, and equations can be obtained from their classical expressions by simply replacing the appropriate operators. Both time–dependent and time–independent Schr¨odinger equations cannot be relativistic, since they were derived with the assumption that the kinetic energy is given by 1 T = (p2 + p2 + p2)(4) 2m x y z However, in relativistic quantum mechanics the kinetic energy is no longer given by this expression, since the mass of a particle is a variable of the speed. The relativistic kinetic energy operator is different from 231 ERKOC¸ the nonrelativistic quantum mechanical kinetic energy term. The potential energy term is also different in relativistic picture. The Schr¨odinger equation is not Lorentz–invariant. Any physically acceptable relativistic wave equation must treat the spatial (space) and temporal (time) coordinates on equal footing, and must be Lorentz–invariant [1]. 2. The Klein–Gordon Equation In 1926 several physicists proposed such equations to treat relativistic quantum–mechanical problems. The energy of a free particle is given by 2 2 2 4 E = p c + m0c (5) →− ∇ → ∂ Substituting p i¯h ,andE i¯h ∂t and then operating on a wavefunction ψ(x, y, z, t), we obtain ∂ψ i¯h = −¯h2c2∇2 + m2c4ψ (6) ∂t 0 This form has some difficulties in solutions, the presence of a square–root term with the Laplacian operator poses some problems. A logical starting point is to use the square of the relativistic energy expression, 2 2 2 2 4 E = p c + m0c (7) This was done by Klein and Gordon [2], independently among several others, and hence came to be known as the Klein–Gordon equation [3,4], ∂2ψ −¯h2 =(−¯h2c2∇2 + m2c4)ψ (8) ∂t2 0 This equation can be expressed in a more convenient form by the introduction of the operator 1 ∂2 ∂2 ∂2 ∂2 ✷ = − − − (9) c2 ∂t2 ∂x2 ∂y2 ∂z2 Thewaveequationthencanbewrittenas 2✷ 2 2 (¯h + m0c )ψ(x, y, z, t) = 0 (10) or, in relativistic units (¯h =1,c =1) ✷ 2 ( + m0)ψ(x, y, z, t) = 0 (11) This equation is known as the Klein–Gordon equation for a free particle in the absence of any electromagnetic field. The solution of this equation yields a relativistic scalar wavefunction for a free particle. Klein–Gordon equation is Lorentz–invariant. For a free particle the solution is in plane–wave form (if ψ depends only on x and t): i (p t−p·x) ψ(x, t)=e h¯ 0 (12) ∓ 2 2 2 4 cp0 = E = c p + m0c (13) The Klein–Gordon equation for a charged particle moving in a static Coulomb field, represented by the wave function − ψ(r,t)=u(r)e iEt (14) is in the form ∇2 2 − − 2 u =[m0 (E eϕ) ]u (15) Taking Ze2 e2 1 u(r)=R(r)Y (θ, φ) ,eϕ(r)=− ,α= ≈ (16) lm 4πr 4π 137 232 ERKOC¸ 1 r = Zeα , ρ = ,E=(m2 − E2)λ2 > 0 λ λ 0 the differential equation becomes d2 2 d Z l(l +1)− Z2α2 + − + − U(ρ) = 0 (17) dρ2 ρ dρ ρ ρ2 where U(ρ)=R(r). The energy expression becomes −1/2 2 2 Z α E = m0 1+ (18) 1 1 2 − 2 2 2 [nr + 2 + (l + 2 ) Z α ] The Klein–Gordon equation for the Coulomb field requires that 1 (Zα)2 ≤ (l + )2 (19) 2 This means for the minimum value of Z, which occurs when l =0, 1 Z> or Z>69. (20) 2α That is if Z>69, the energy eigenvalue can become complex for the s orbitals, and thus the solution of the Klein–Gordon equation suggests that an atom cannot exist with Z>69, as this would lead to complex energy eigenvalues. Therefore, it is clear that the Klein–Gordon equation does not yield satisfactory results for an electron in a Coulomb field. The problem is that an electron is a spin 1/2 particle; one must treat the angular momentum as a four–vector. Thus, the wave function of any spin particle, in general, cannot be described by a simple scalar wave function, one needs to seek a vectorial solution for the wave function in relativistic quantum mechanics of any particle possessing spin. 3. The Dirac Equation Dirac in 1928 proposed a relativistic quantum–mechanical equation, now well known as the Dirac equa- tion. Dirac proposed a first–order differential equation in both temporal and spatial coordinates. Dirac assumed that the relativistic wave function ψ consists of n components for a general positive integer n; subsequently, it will be established that n is 4 [1]. Thus ψ is treated as a n–dimensional column vector with components ψ1,ψ2, ···,ψn. The general first–order differential equation in both spatial and temporal coordinates looks like: 3 n n 1 ∂ψ ∂ψ im c l + αk i + 0 β ψ = 0 (21) c ∂t li ∂xk ¯h li i k=1 i=1 i=1 where l =1, 2, ···,n. α and β are dimensionless constants. The Dirac equation is obtained once n and the matrices α and β are determined. This is accomplished by stipulating that (a) the continuity equation for the charge density current be satisfied and (b) the free– particle solution satisfies the free–particle Klein–Gordon equation. The matrices α and β are constructed as follows: 0 σ I 0 αk = k ,β= (22) σk 0 0 I where 10 01 0 i 10 I = ,σ= ,σ= ,σ= (23) 01 1 10 2 −i 0 3 0 −1 σk are Pauli matrices. The Dirac equation for a free particle in the absence of any external field takes the form [1] 1 ∂ψ im c +(α ·∇)ψ + 0 βψ = 0 (24) c ∂t ¯h 233 ERKOC¸ →− ∇ → ∂ or (considering p i¯h , E i¯h ∂t 2 [c(α · p)+βm0c ]ψ = Eψ (25) Therefore, we derive the Dirac Hamiltonian for a free particle in the absence of any external field as 2 H = c(α · p)+βm0c (26) The Dirac equation, as well as the Dirac Hamiltonian, is a 4 × 4 matrix equation. Consequently, the wavefunction is also a column vector of length 4. This four–component wave function is called a spinor. Using the relations γk = −iβαk, k =1, 2, 3, and γ4 = β, Dirac equation can be simplified further, (in relativistic units, that ish ¯ =1,c =1) ∂ (γµ + m0)ψ =0 ,µ=1, 2, 3, 4 (27) ∂xµ The explicit form of the four equations in terms of spinors: ∂ ∂ ∂ ∂ ( + im )ψ +( − i )ψ + ψ = 0 (28) ∂t 0 1 ∂x ∂y 4 ∂z 3 ∂ ∂ ∂ ∂ ( + im )ψ +( + i )ψ − ψ = 0 (29) ∂t 0 2 ∂x ∂y 3 ∂z 4 ∂ ∂ ∂ ∂ ( − im )ψ +( − i )ψ + ψ = 0 (30) ∂t 0 3 ∂x ∂y 2 ∂z 1 ∂ ∂ ∂ ∂ ( − im )ψ +( + i )ψ − ψ = 0 (31) ∂t 0 4 ∂x ∂y 1 ∂z 2 For free electron we seek a plane–wave solution for each component. Thus the following form for the components is taken: i(p·r−Et) ψµ = Aµe (32) The expression for energy, E, takes the following form: 2 − 2 − 2 2 2 2 [E m0 (p1 + p2 + p3)] = 0 (33) or ∓ 2 2 2 2 E∓ = p1 + p2 + p3 + m0 (34) The Dirac equation for a charged particle in an electromagnetic field is given as (γµΠµ − im0)ψ = 0 (35) This equation explicitly splits into the following four equations in terms of the four components, ∂ (−i − eϕ + m )ψ +Π−ψ +Π ψ = 0 (36) ∂t 0 1 4 3 3 ∂ (−i − eϕ + m )ψ +Π ψ − Π ψ = 0 (37) ∂t 0 2 + 3 3 4 ∂ (−i − eϕ + m )ψ +Π−ψ +Π ψ = 0 (38) ∂t 0 3 2 3 1 ∂ (−i − eϕ + m )ψ +Π ψ − Π ψ = 0 (39) ∂t 0 4 + 1 3 2 where Π∓ =Π1 ∓ iΠ2 (40) ∂ Πk = −i + eAk (k =1, 2, 3) (spatial conjugate) (41) ∂xk ∂ Π = − + ieϕ (temporal conjugate) (42) 4 ∂t 234 ERKOC¸ 4.
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