Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2017), 038, 15 pages Mendeleev Table: a Proof of Madelung Ruleand Atomic Tietz Potential
Eugene D. BELOKOLOSDepartment of Theoretical Physics, Institute of Magnetism,National Academy of Sciences of Ukraine, 36-b Vernadsky Blvd., Kyiv, 252142, Ukraine
E-mail: [email protected]
Received February 27, 2017, in final form May 22, 2017; Published online June 07, 2017https://doi.org/10.3842/SIGMA.2017.038
Abstract.
We prove that a neutral atom in mean-field approximation has O(4) symmetryand this fact explains the empirical [ n + l, n ]-rule or Madelung rule which describes effectivelyperiods, structure and other properties of the Mendeleev table of chemical elements. Key words:
Madelung rule; Mendeleev periodic system of elements; Tietz potential
In 1869 D.I. Mendeleev discovered a periodic dependence of chemical element properties on Z with periods2 , , , , , , . . . , (1.1)where Z is a number of electrons in the chemical element atom .With creation of quantum mechanics physicists tried to explain the Mendeleev empirical lawin terms of the one-particle quantum numbers n, l, m, σ, where n = n r + l +1 is a principal quantum number, n r is a radial quantum number, l is an orbitalquantum number, m is a magnetic quantum number, − l ≤ m ≤ l , σ = ± / n only and are degenerate in other quantum numbers: l , 0 ≤ l ≤ n − m , − l ≤ m ≤ l ; σ , σ = ± /
2. It isa so called [ n, l ]-rule. According to it the energy spectrum of atom consists of electron shells,enumerated by the principal quantum number n and having the degeneracy N n = n (cid:88) l =0 l + 1) = 2 n . This formula gives the following set of periods2 , , , , . . . . (1.2)Comparing the period sequences (1.1) and (1.2), we can explain only the first two periods ofthe Mendeleev table. Although we see a remarkable fact that (1) the period lengths have cardi-nalities that correspond to the Hydrogen degeneracy dimensions, and (2) the same cardinalitiesalways occur in pairs, except for the very first one. In this paper we show that these similaritiesare not accidental. See https://en.wikipedia.org/wiki/Periodic_table . a r X i v : . [ phy s i c s . a t o m - ph ] J un E.D. Belokolos n + l, n ]-rule We get an exact expression for the Mendeleev periods and other properties of the Mendeleevtable with the Madelung [ n + l, n ]-rule [22]).The [ n + l, n ]-rule asserts: with growth of atomic charge Z the electrons fill up in atomconsecutively the one-particle states with the least possible value of the quantum number n + l ;and, for a given value n + l , the electrons fill up states with the least possible value of thequantum number n .Here it is reasonable to introduce the quantum number M = n + l = n r + 2 l + 1 . The [ n + l, n ]-rule is in fact an algorithm for consecutive building-up of atoms . It describes thestates for each of about 5000 electrons of the atoms to corresponding elements of the periodicsystem (indeed, now we have 118 chemical elements, and for approximately 100 elements weknow every state of electron configuration, that is (cid:80) Z = 5050 states) and predicts correctly thereal electron configurations of all elements of periodic system with small number of exclusions.There exist only 19 elements (Cr, Cu, Nb, Mo, Ru, Rh, Pd, Ag, La, Ce, Gd, Pt, Au, Ac, Th,Pa, U, Np, Cm), whose electron configurations differ from the configurations predicted withthe [ n + l, n ]-rule [24].The [ n + l, n ]-rule allows to obtain relations between the order number Z of chemical elementand quantum numbers M , n , l of the appropriate electron configuration.Let us designate the order number Z of chemical element in which electrons from the ( n + l )-subgroups, ( n, l )-subgroups, n -subgroups, l -subgroups or n r -subgroups appear for the first timeby symbols Z n + l , Z n,l , Z n , Z l , Z n r . Let us designate an order number Z of a chemical element in which the ( n + l )-subgroups, ( n, l )-subgroups, n -subgroups, l -subgroups or n r -subgroups are filled up completely for the first timeby symbols n + l Z, n,l Z, n Z, l Z, n r Z. Then the [ n + l, n ]-rule with help of four actions of arithmetic and well known formulas k = n (cid:88) k =1 k = (1 / n ( n + 1) , k = n (cid:88) k =1 k = (1 / n ( n + 1)(2 n + 1) , leads to the Klechkovski–Hakala formulas [11, 15] Z n + l = K ( n + l ) + 1 , n + l Z = K ( n + l + 1) ,Z n,l = K ( n + l + 1) − l + 1) + 1 , n,l Z = K ( n + l + 1) − l ,Z n = K ( n + 1) − , n Z = K (2 n ) − n − = (1 / (cid:2) (2 n − + 11(2 n − (cid:3) ,Z l = K (2 l + 1) + 1 = (1 / (cid:2) (2 l + 1) + (5 − l ) (cid:3) ,Z n r = K ( n r + 2) − . (2.1) See https://en.wikipedia.org/wiki/Aufbau_principle . endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 3Here K ( x ) is the following function K ( x ) = (1 / x (cid:2) x + 2 − µ ( x ) (cid:3) , x ∈ N ,µ ( x ) = x mod (2) = (cid:40) , x is odd , , x is even . Relations for Z n + l and n + l Z are exact. Other relations are exact up to exclusions pointedout above. For example, for l = 0 , , . . . , we have Z l = 1 , , , , . . . , that is, according to theMadelung rule, the onset of the 4f block starts with La ( Z = 57). But, according to the IUPACdata on electron configurations in atoms, the onset of the 4f block starts with Ce ( Z = 58).The [ n + l, l ]-rule defines essential characteristics of the Mendeleev periodic system.In the Mendeleev periodic system, the table rows are enumerated by the Mendeleev number M which is a linear function of the Madelung number M [15]: M = M − δ l, = n + l − δ l, , where δ j,k is the Kronecker delta.It is easy to show that the number of elements in the Mendeleev M -th period of the periodicsystem according to the [ n + l, n ]-rule is equal to L M = K ( M + 2) − K ( M + 1) = 2 (cid:18)(cid:20) M (cid:21) + 1 (cid:19) , where [ x ] is an integer part of the real number x . Numbers L M , M = 1 , , . . . form the sequence2 , , , , , , , . . . , which coincides with empirical lengths (1.1) of the periods of the systemof elements.According to the [ n + l, n ]-rule, the number Z M for the initial element of the Mendeleev M -thperiod is equal to Z M = K ( M + 1) − , and the number M Z for the final element of the Mendeleev M -th period is equal to M Z = K ( M + 2) − . The sequence Z M = 1 , , , , , , , . . . corresponds to alkaline metals, and the sequence M Z = 2 , , , , , , . . . corresponds to noble gases.Thus, the empirical [ n + l, n ]-rule is a very efficient method to explain periodic table andatomic properties. Research to justify it continues up to now (see, e.g., [2, 13, 14, 26, 27]. Butafter 80 years of studies we have not yet good understanding for it. Standard textbooks onquantum mechanics even do not mention this rule.In the present paper we give a theoretical basis for the [ n + l, n ]-rule. A preliminary versionof this paper was published in 2002 [6].Further we use everywhere the atomic units (cid:126) = e = m = 1, where (cid:126) is the reduced Plankconstant, − e is the electron charge, and m is the electron mass. See https://iupac.org/what-we-do/periodic-table-of-elements/ . E.D. Belokolos
The Hamiltonian of a free neutral atom is H A = Z (cid:88) k =1 (cid:18) −
12 ∆ k − Z(cid:126)r k (cid:19) + 12 Z (cid:88) i,k =1 (cid:126)r ik = Z (cid:88) k =1 (cid:18) −
12 ∆ k + v ( (cid:126)r k ) (cid:19) ,v ( (cid:126)r k ) = − Z(cid:126)r k + 12 Z (cid:88) i =1 (cid:126)r ik , where v ( (cid:126)r ) describes the electron interactions with the atomic nucleus and other electrons.Since it is good approximation to describe the electrons in an atom in terms of the electronconfiguration, i.e., the electron distribution on one-particle states, n, l, m, σ, therefore we may consider the electron interaction v ( (cid:126)r ) in the mean-field approximation withthe Hamiltonian H = Z (cid:88) k =1 (cid:18) −
12 ∆ k + V ( (cid:126)r k ) (cid:19) , where V ( (cid:126)r ) is the mean-field atomic potential. Since we consider a one-particle angular momen-tum, quantum number l as a good quantum number, this potential has to be central, V ( (cid:126)r ) = V ( r ) . For the Schr¨odinger equation H Ψ = E Ψwe look for the ground state solution in the Slater determinant formΨ = det || ψ j ( r k ) || , where one-particle wave function ψ k ( r ) satisfies the one-particle Schr¨odinger equation describingan electron in a 3-dimensional central potential V ( r ), (cid:18) −
12 ∆ + V ( r ) (cid:19) ψ j ( r ) = E j ψ j ( r ) . In the semiclassical approximation, solutions of this equation have the form ψ ( r ) = exp( iS ( r )) , where S ( r ) is the classical action. Since in the case under consideration we have three integrals of motion (energy E , angularmomentum L , and its projection L z ) our problem is integrable by quadratures due to theArnold–Liouville theorem, and the radial action looks as follows S ( r ) = 1 π (cid:90) r (cid:20) E − V ( r )) − I θ r (cid:21) / d r, endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 5where the radial action variable is I r = 1 π (cid:90) r max r min (cid:20) E − V ( r )) − I θ r (cid:21) / d r. In the semiclassical approximation we should change action variables by quantum numbers, I r = n r ( E, l ) , I θ = l. In this way we come to the Bohr–Sommerfeld quantization rule n r ( E, l ) = 1 π (cid:90) r + ( E,l ) r − ( E,l ) (cid:20) E − V ( r ) − l r (cid:21) / d r. Let us set in the latter expression E = 0, then n r (0 , l ) = √ π (cid:90) r + (0 ,l ) r − (0 ,l ) (cid:20) − V ( r ) − l r (cid:21) / d r. Thus the number of bound states N in the atom is N = l max (cid:88) l =0 n r (0 , l )2(2 l + 1) . Approximately a value N looks as follows N (cid:39) (cid:90) l max l =0 n r (0 , l )2(2 l + 1)d l = √ π (cid:90) r + r − (cid:90) l max l =0 (cid:20) − V ( r ) − l r (cid:21) / l + 1)d l d r = √ π (cid:90) r + r − [ − V ( r )] / r d r = 2 √ π (cid:90) r + r − [ − V ( r )] / πr d r = (cid:90) r + r − ρ ( r )4 πr d r. This is the well-known asymptotic formula [29] for the number of bound states in a centralpotential with the density of bound states ρ ( r ) equal to ρ ( r ) = 2 √ π [ − V ( r )] / . The electrostatic potential of atomic electrons φ ( r ) = − V ( r ) satisfies the Poisson equation∆ φ ( r ) = − πρ , and therefore∆ φ ( r ) = 8 √ π φ / ( r ) ,φ ( r ) r = Z, r → , φ ( r ) = 0 , r → ∞ . We can present the Thomas–Fermi potential φ ( r ) in such a way φ ( r ) = Zr χ ( x ) , x = rR , R = bZ − / , b = 12 (cid:18) π (cid:19) / (cid:39) . . For large Z the ground state energy E TF ( Z ) of the Thomas–Fermi atom is the asymptoticsfor a ground state energy E HF ( Z ) of the Hartree–Fock atom [20, 21]:lim Z →∞ E HF ( Z ) /E TF ( Z ) = 1 . E.D. BelokolosThe Thomas–Fermi potential does not depend on quantum numbers although naturally itshould do. Nevertheless, let us calculate the Z l for the Thomas–Fermi potential [8].The effective potential u l ( r ) is u l ( r ) = − φ ( r ) + ( l + 1 / r = − ( l + 1 / r [ ζ l xχ ( x ) − , where ζ l = 2 ZR ( l + 1 / = 2 b Z / ( l + 1 / . Conditions for the first appearance of the energy level with certain l are u l ( r ) = 0 , u (cid:48) l ( r ) = 0or, which is the same, u l ( x ) = 0 , u (cid:48) l ( x ) = 0 . It is equivalent to equations ζ l xχ ( x ) − , dd x ( xχ ( x )) = 0 , which have a solution x (cid:39) . , χ ( x ) (cid:39) . , x χ ( x ) (cid:39) . ,χ (cid:48) ( x ) (cid:39) − . , ζ l = [ x χ ( x )] − (cid:39) . . This means that Z TF l (2 l + 1) = (cid:18) ζ l b (cid:19) / = 0 . , and hence Z TF l (cid:39) . l + 1) . In the general case Z T Fl is not integer and we have to write Z TF l = (cid:12)(cid:12) . l + 1) (cid:12)(cid:12) , where | x | means the integer which is the closest to the real x .If we change in this formula the coefficient 0 .
155 to 0 . (cid:39) / Z T Fl with that in the Mendeleev table (see [12] and [17, Section 73]). In this case ther.h.s. of the formula will coincide with the first summand of the Klechkovski–Hakala expressionfor Z l (2.1) obtained with the [ n + l ]-rule.This example shows that the Thomas–Fermi theory yields only some approximation to the[ n + l, n ]-rule.According to the above discussion the electrons in the atom in the mean-field approximationinteract by means of central atomic potential, having geometrical O(3) symmetry. The Thomas–Fermi theory takes into account the O(3) symmetry of the atomic Hamiltonian with centralpotential. We do the next step. The Hamiltonian with any central potential has dynamical O(4)symmetry [4, 10, 25], similar to the Hydrogen atom [5, 9, 28]. Therefore, the atomic Hamiltonianmust have an additional integral of motion in involution with respect to integrals of motioncorresponding to O(3) symmetry. Further we shall show that such additional integral of motionin involution does exist for the atomic Hamiltonian and leads to [ n + l, n ]-rule.endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 7 We begin with certain basic facts on symmetry and integrability in the classical Hamiltoniansystems (see [3, Sections 49–51], [16, Section 52] and [33]).Let us assume that on a symplectic 2 n -dimensional manifold there are n functions in involu-tion F , . . . , F n , { F i , F j } = 0 , i, j = 1 , . . . , n. Consider a level set of functions F i M f = { x : F i ( x ) = f i , i = 1 , . . . , n } . Suppose that n functions F i are independent on M f (i.e., the n dF i are linearly inde-pendent in every point of M f ).Then1. M f is a smooth manifold, invariant with respect to the phase flow with the Hamiltonian H = F .2. If the manifold M f is compact and connected, then it is diffeomorphic to n -dimensionaltorus T n = { ( φ , . . . , φ n ) (mod 2 π ) } .
3. Phase flow with the Hamiltonian H defines on M f a quasi-periodic movement with angularvariables ( φ , . . . , φ n ),d φ i d t = ω i ( f ) , φ i ( t ) = φ i (0) + ω i t, i = 1 , . . . , n. Instead of functions F = ( F , . . . , F n ) it is possible to define new functions I = ( I ( F ) , . . . , I n ( F )) which are called action variables and which together with angle variables form inthe neighborhood of manifold M f the canonical system of action-angle coordinates.4. Canonical equations with the Hamiltonian function are integrable in quadratures.5. If frequencies ω = ( ω , . . . , ω n ) are degenerated, i.e., if there exists such an integer-valuedvector k = ( k , . . . , k n ) ∈ Z n that( k, ω ) = 0 , then there appears one more single-valued function F n +1 which is in involution with func-tions F , . . . , F n .Now let us go back to our problem: an electron in the central atomic potential. In this casethe 3-dimensional electron movement is reduced to 2-dimensional one in the plane perpendicularto the orbital momentum vector. We describe this movement in the semiclassical approximationby the action S = (cid:90) r (cid:115) E − V ( r )] − I θ r d r with radial ( r, I r ) and orbital ( θ, I θ ) action-angle variables, where I r = 1 π (cid:90) r max r min (cid:115) E − V ( r )] − I θ r d r. E.D. BelokolosAnd so in our problem we have a quasi-periodic movement on 2-dimensional torus which isdescribed by the Fourier series G ( t ) = (cid:88) l ∈ Z (cid:88) l ∈ Z G l ,l exp( l φ r + l φ θ ) = (cid:88) l ∈ Z (cid:88) l ∈ Z G l ,l exp[( l ω r + l ω θ ) t ]with 2 basic frequencies ω r = ∂E∂I r , ω θ = ∂E∂I θ . In the general case these frequencies are independent. But under certain circumstances, as wehave, when new additional integral appears, they may become degenerate (or commensurate, orresonance), qω r = pω θ , q, p ∈ Z , g . c . d . ( p, q ) = 1 , where g . c . d . ( p, q ) means the greatest common divisor of the integers p and q . This conditionleads to important consequences.1. Since the latter relation is equivalent to q ∂E∂I r = p ∂E∂I θ it means that energy E depends on pI r + qI θ : E = E ( pI r + qI θ ) . And due to the Bohr–Sommerfeld semiclassical quantization rule I r → n r , I θ → l, we have E = E ( pn r + ql ) .
2. We have the additional independent integral of motion.3. The canonical variables of the problem are separated in several systems of coordinates.For example, for the Kepler–Coulomb potential 1 /r we have p = q = 1, energy E depends onprincipal quantum number n = n r + l + 1, the canonical variables of the problem are separatedin polar and parabolic coordinates.Let us study the similar situation for the atomic potential. Consider a system of equations I r ( E ) = 1 π (cid:90) r + ( E ) r − ( E ) (cid:20) E − V ( r ) − I θ r (cid:21) / d r,M p,q = pI r + qI θ , q, p ∈ Z , g . c . d . ( p, q ) = 1 , the first of which is the standard relation between action variables I r , I θ , and energy E incentral potential and the second one is relation between I r , I θ arising as result of frequenciesdegeneracy.endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 9We shall study equations at the energy E = 0 at which new bound states appear fromcontinuous spectrum, when Z grows. Then differentiating equations with respect to I θ we cometo the equation πα = (cid:90) r + r − (cid:2) − V ( r ) r − I θ (cid:3) − / I θ d rr , where α = − ∂I r ∂I θ = qp ∈ Q , q, p ∈ Z , g . c . d . ( p, q ) = 1 . From this equation we deduce an expression for atomic potential V ( r ). A similar integralequation arising in problem for generalized tautochrone (isochrone) curve was solved by Abel [1].We shall use his method in the form presented in [16, Section 12]. Theorem 1.
Equation πα = (cid:90) r + r − (cid:2) − V ( r ) r − L (cid:3) − / L d rr (3.1) has a solution V α,β,R ( r ) = − βr [( r/R ) /α + ( R/r ) /α ] , where β and R are certain constants, and its deformations.If the potential V α,β,R ( r ) coincides at small r with the Kepler–Coulomb potential V α,β,R = − Z/r, r → , then we have α = q/p = 2 , β = ZR, and the potential takes the following form V ( r ) ≡ V ,ZR,R ( r ) = − Zr (1 + ( r/R )) . Remark 1.
According to the theorem in the neighborhood E (cid:39) E = E ( n + l )and this fact proves the first part of the [ n + l, n ]-rule. A second part of the [ n + l, n ]-rule isa consequence of the oscillation theorem. Proof .
Let us rewrite equation (3.1) in the form πα = (cid:90) r + r − (cid:2) w ( x ) − L (cid:3) − / L d x, where x = ln( r/R ) , r = R exp( x ) ,w ( x ) = − V ( r ) r at r = R exp( x ) ,w = max w ( x ) , R is a parameter. We assume that w ( x ) is the one-well potential andtherefore the inverse function is two-valued, i.e., the values w are reached in two points x − ( w )and x + ( w ). We shall assume also that x − ( w ) ≤ x + ( w ) and at w we have x − ( w ) = x + ( w ).As a result we obtain πα = (cid:90) w L (cid:0) w − L (cid:1) − / (cid:18) d x + d w − d x − d w (cid:19) L d w. Multiplying this equality by (cid:0) L − w (cid:1) − / L and integrating from ( w ) / to ( w ) / weget 2 πα (cid:90) ( w ) / ( w ) / (cid:0) L − w (cid:1) − / d L = (cid:90) ( w ) / ( w ) / L d L (cid:90) w L (cid:18) d x + d w − d x − d w (cid:19) (cid:2)(cid:0) w − L (cid:1)(cid:0) L − w (cid:1)(cid:3) − / d w = (cid:90) w w d w (cid:18) d x + d w − d x − d w (cid:19) (cid:90) ( w ) / ( w ) / (cid:2)(cid:0) w − L (cid:1)(cid:0) L − w (cid:1)(cid:3) − / L d L. Since (cid:90) ( w ) / ( w ) / (cid:0) L − w (cid:1) − / d L = Arcosh( w /w ) / , (cid:90) ( w ) / ( w ) / (cid:2)(cid:0) w − L (cid:1)(cid:0) L − w (cid:1)(cid:3) − / L d L = π, this equality acquires the following form2 α Arcosh( w /w ) / = ( x + ( w ) − x − ( w )) (cid:12)(cid:12)(cid:12) w w . Taking into account that x − ( w ) = x + ( w ) and setting w = w we come to the equality x + ( w ) − x − ( w ) = 2 α Arcosh( w /w ) / . This equality defines only the difference x − ( w ) − x + ( w ) of two functions x − ( w ) and x + ( w ),any of which remains actually undefined. It means that there exists the infinite set of potentialswhich satisfy the latter equation and differ by deformations which do not change the differenceof two values of x corresponding to one value of w . Among these potentials there is a symmetricpotential with the property x + ( w ) = − x − ( w ) ≡ x ( w ). For this potential in this case w ( x ) = w / cosh ( x/α ) , or, in the previous notations, V α,β,R ( r ) = − βr [( r/R ) /α + ( R/r ) /α ] , where β = 2 w . These potentials are degenerate for arbitrary value of the parameter α . In orderfor the degenerate potential to coincide with Coulomb potential at small rV α,β,R = − Z/r, r → , we must have α = q/p = 2 , β = ZR.
Then this potential takes the following form V ( r ) ≡ V ,ZR,R ( r ) = − Zr (1 + ( r/R )) . (cid:4) endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 11In this potential the total number of bound states N is equal to the total number of elect-rons Z if N = 8 √ π (cid:90) [ − V ( r )] / d r = (cid:0) / R (cid:1) = Z, i.e., R = (9 / Z ) / (cid:39) . Z − / , and we have finally V ( r | Z ) = − Zr (1 + ( r/R ( Z ))) . For the given electron configuration we must use the Klechkovski–Hakala formulas for Z . Inthis case the potential will depend on quantum numbers.We can present this potential as the sum V ( x ) = − ( Z/R ) x (1 + x ) = − ZR (cid:20) x − x + 1) − x + 1) (cid:21) , x = rR , where the first summand describes the Coulomb attraction of the atomic nucleus for the singleelectron and two other summands describe a nucleus screening by the other electrons.T. Tietz [30] proposed the potential V ( r ) = − Zr [1 + ( r/R )] as a good rational approximation to the Thomas–Fermi potential and used it for calculation ofvarious atomic properties and explanation of the periodic system elements [31] (see also [34]).Further we shall call this potential as the Tietz atomic potential.Due to the proximity of the Tietz and Thomas–Fermi potentials it is very likely that atlarge Z a ground state energy of the Tietz atom, E T ( Z ), is an asymptotics for the ground stateenergy of the Hartree–Fock atom, E HF ( Z ):lim Z →∞ E HF ( Z ) /E T ( Z ) = 1 . Yu.N. Demkov and V.N. Ostrovski [7] pointed out that this potential is a particular case ofa so called focussing (in other words “degenerate”) potentials studied in connection to certainproblems of optics by J.C. Maxwell [23] and V. Lenz [19].The Tietz potential is a rational function, and thus we can easily do various calculations withit. For example, we can calculate the atomic spectrum.
The Bohr–Sommerfeld semiclassical condition of quantization for the Tietz atomic potential is n r = 1 π (cid:90) r + r − (cid:20) E + 2 Zr (1 + ( r/R )) − ( l + 1 / r (cid:21) / d r. In the scaled quantum numbers x = rR , (cid:15) = 2 ER ( l + 1 / , ν r = n r l + 1 / , η l = 2 ZR ( l + 1 / ν r = 1 π (cid:90) x + x − (cid:20) (cid:15) + η l x (1 + x ) − x (cid:21) / d x = 1 π (cid:90) x + x − (cid:112) P ( x ) x (1 + x ) d x, where P ( x ) is a 4-th degree polynomial P ( x ) = (cid:15)x (1 + x ) + η l x − (1 + x ) , and boundaries of integration are real non-negative zeros of this polynomial. Therefore scaledradial quantum number ν r is a period of the elliptic integral with scaled energy (cid:15) and scaledcharge η l as the parameters. We can obtain an atomic spectrum in the semiclassical approxi-mation (cid:15) = f ( ν r , η l ) , by means of inversion of the elliptic integral.Let us study this problem in the framework of perturbation theory in the energy (cid:15) . Wepresent the Bohr–Sommerfeld semiclassical equation in the form ν r = 1 π (cid:90) x + x − (cid:20) (cid:15) + η l x − (1 + x ) x (1 + x ) (cid:21) / d x = 1 π (cid:90) x + x − (cid:20) (cid:15) + ( x + − x )( x − x − ) x (1 + x ) (cid:21) / d x, where η l x − (1 + x ) = ( x + − x )( x − x − ) ,x ± − ( η l − x ± + 1 = 0 ,x ± = ( η l − ± (cid:112) ( η l − −
42 = ( η l − ± (cid:112) η l ( η l − , − ( η l − > (cid:15) > − η l ( η l − . At the first approximation we have ν r (cid:39) J + (cid:15)J = 1 π (cid:90) x + x − [( x + − x )( x − x − )] / d xx (1 + x ) + (cid:15) π (cid:90) x + x − x (1 + x )[( x + − x )( x − x − )] / d x, where J = 1 π (cid:90) x + x − [( x + − x )( x − x − )] / d xx (1 + x ) = √ η l − ,J = 12 π (cid:90) x + x − x (1 + x )[( x + − x )( x − x − )] / d x = 116 (cid:0) η l − η l (cid:1) . The energy spectrum in the semiclassical theory is (cid:15) n,l = − J − ν r J = − √ η l − − ν r (cid:0) η l − η l (cid:1) = −
16 ( l + (1 / √ η l − l − − n r ( l + (1 / (cid:0) η l − η l (cid:1) = −
16 ( l + (1 / √ η l − ( n + l )( l + (1 / (cid:0) η l − η l (cid:1) = − √ ZR − M ( l + (1 / (cid:0) η l − η l (cid:1) , where we have used the Madelung number M = n + l .endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 13Since (cid:15) M,l = 2 E M,l R ( l + (1 / , we have the following expression for the atomic spectrum in the atomic units E M,l = − √ ZR − M )( l + (1 / (cid:0) η l − η l (cid:1) R , η l = 2 ZR ( l + (1 / . If √ ZR = M then E M,l = 0, and all states (
M, l ) are degenerate with respect to l . Thus at energy E = 0 thereappears the full set of energy levels ( n, l ) with M = n + l . Since R = (9 / Z ) / the equality M = √ ZR is equivalent to Z = (1 / M . We can write down a complete perturbation series ν r = ∞ (cid:88) k =0 (cid:15) k J k . (4.1)Since all integrals J k = c k π (cid:90) x + x − (cid:26) x (1 + x ) [( x + − x )( x − x − )] (cid:27) k − (1 / d x,c k = Γ(3 / k + 1)Γ((3 / − k ) , k ≥ , are divergent we should take their regularized values. i.e., valeur principale. Inverting series (4.1)by means of the B¨urmann–Lagrange theorem we can obtain an expression for atomic spectrumin the semiclassical approximation (cid:15) = f ( ν r , η l ) . We plans to compare this atomic spectrum (both eigenvalues and eigenfunctions) with thosepresented in articles [18, 32].
For the Mendeleev periodic system of elements we have proved the empirical ( n + l, n )-rule whichexplains very efficiently the structure and properties of chemical elements. In order to provethis rule we are forced to build for the Hamiltonian describing an electron in central atomicpotential one more integral of motion in involution in addition to energy and orbital integral.In this case the atomic potential appears to be the Tietz potential. For the Tietz potential wehave calculated the atomic energy spectrum in the semiclassical approximation.In addition, we are going to study spectrum of the Schr¨odinger operator with the Tietzpotential, when the radial part of wave function satisfies the confluent Heun equation andcompare energy levels, obtained in such a way, with the NIST Atomic Spectra Database.4 E.D. BelokolosFor the atom in the paper we have studied the non-relativistic atomic Hamiltonian. We planto consider also the relativistic Dirac Hamiltonian.It is interesting also to study isospectral many-well deformation of the Tietz potential.Questions studied in the paper are related only to the ground state of atoms. But accordingto experimental data the weakly excited atomic states also follow to [ n + l, n ]-rule [15]. It maybe of interest for physical chemistry, especially for understanding chemical reactions rules.We hope to do that in the near future. References [1] Abel N.H., Solution de quelques probl´emes `a l’aide d’int´egrales d´efinies, in Œuvres completes de Niels HenrikAbel, Editors L. Sylow, S. Lie, Christiania, Norway, 1881, 11–27.[2] Allen L.C., Knight E.T., The L¨owdin challenge: origin of the n + l, n (Madelung) rule for filling the orbitalconfigurations of the periodic table, Int. J. Quantum Chem. (2002), 80–88.[3] Arnol’d V.I., Mathematical methods of classical mechanics, 3rd ed., Nauka, Moscow, 1989.[4] Bacry H., Ruegg H., Souriau J.M., Dynamical groups and spherical potentials in classical mechanics, Comm.Math. Phys. (1966), 323–333.[5] Bargmann V., Zur Theorie des Wasserstoffatoms. Bemerkungen zur gleichnamigen Arbeit von V. Fock, Z. Phys. (1936), 576–582.[6] Belokolos E.D., The integrability and the structure of atom, J. Math. Phys. Anal. Geometry (2002),339–351.[7] Demkov Y.N., Ostrovskii V.N., Internal symmetry of the Maxwell “fish eye” problem and the Fock groupfor the hydrogen atom, Sov. Phys. JETP (1971), 1083–1087.[8] Fermi E., ¨Uber die Anwendung der statistischen Methode auf die Probleme des Atombaues, in Quantenthe-orie und Chemie, Editor H. Falkenhagen, Leipziger Vortr¨age, S. Hirzel-Verlag, Leipzig, 1928, 95–111.[9] Fock V., Zur Theorie des Wasserstoffatoms, Z. Phys. (1935), 145–154.[10] Fradkin D.M., Existence of the dynamic symmeries O(4) and SU for all classical central potential problems, Progr. Theoret. Phys. (1967), 798–812.[11] Hakala R., The periodic law in mathematical form, J. Phys. Chem. (1952), 178–181.[12] Ivanenko D., Larin S., Theory of periodic system of elements, Dokl. Akad. Nauk USSR (1953), 45–48.[13] Kibler M.R., On a group-theoretical approach to the periodic table of chemical elements, quant-ph/0408104.[14] Kibler M.R., From the Mendeleev periodic table to particle physics and back to the periodic table,quant-ph/0611287.[15] Klechkovski V.M., The distribution of atomic electrons and the filling rule for ( n + l )-groups, Atomizdat,Moscow, 1968.[16] Landau L.D., Lifshits E.M., Theoretical physics, Vol. 1, Mechanics, Nauka, Moscow, 1973.[17] Landau L.D., Lifshits E.M., Theoretical physics, Vol. 3, Quantum mechanics: nonrelativistic theory, Nauka,Moscow, 1974.[18] Latter R., Atomic energy levels for the Thomas–Fermi and Thomas–Fermi–Dirac potentials, Phys. Rev. (1955), 510–519.[19] Lenz W., Zur Theorie der optischen Abbildung, in Probleme der mathematischen Physik, Leipzig, 1928,198–207.[20] Lieb E.H., Thomas–Fermi and related theories of atoms and molecules, Rev. Modern Phys. (1981),603–641.[21] Lieb E.H., Simon B., The Thomas–Fermi theory of atoms, molecules and solids, Adv. Math. (1977),22–116.[22] Madelung E., Die mathematischen Hilfsmittel des Physikers, Die Grundlehren der Mathematischen Wis-senschaften , Vol. 4, Springer-Verlag, Berlin, 1936.[23] Maxwell J.C., Solutions of problems, in The Scientific Papers of James Clerk Maxwell, Editor W.D. Niven,Dover Publications, New York, 1952, 76–79. endeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 15 [24] Meek T.I., Allen L.C., Configuration irregularities: deviations from the Madelung rule and inversion oforbital energy levels,
Chem. Phys. Lett. (2002), 362–364.[25] Mukunda N., Dynamical symmeries and classical mechanics,
Phys. Rev. (1967), 1383–1386.[26] Ostrovsky V.N., Dynamic symmetry of atomic potential,
J. Phys. B: At. Mol. Phys. (1981), 4425–4439.[27] Ostrovsky V.N., What and how physics contributes to understanding the periodic law?, Found. Chem. (2001), 145–182.[28] Pauli W., ¨Uber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Phys. (1926),336–363.[29] Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press,New York – London, 1978.[30] Tietz T., ¨Uber eine Approximation der Fermischen Verteilungsfunktion, Ann. Physik (1955), 186–188.[31] Tietz T., Elektronengruppen im periodischen System der Elemente in der statistischen Theorie des Atoms,
Ann. Physik (1960), 237–240.[32] Tietz T., ¨Uber Eigenwerte und Eigenfunktionen der Schr¨odinger-Gleichung f¨ur das Thomas–Fermische Po-tential,
Acta Phys. Acad. Sci. Hungar. (1960), 391–400.[33] Torrielli A., Classical integrability, J. Phys. A: Math. Theor. (2016), 323001, 31 pages, arXiv:1606.02946.[34] Wong D.P., Theoretical justification of Madelung’s rule, J. Chem. Educ.56