Electric Dipole Moment of Atomic Yb Arising from an Electron-Nucleus Interaction

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1 Electric Dipole Moment of Atomic Yb Arising from an Electron-Nucleus Interaction Angom Dilip Singh, Bhanu Pratap Das, M. K. Samal Non Accelerator Particle Physics Group, Indian Institute of Astrophysics, Sarjapur Road, Bangalore Warren F. Perger Michigan Tech University, Electrical Engineering and Physics Departments, 1400 Townsend Drive, Houghton, MI The existence of intrinsic electric dipole momentedm of an atom implies simultaneous parity and time reversal violation. Within the standard modelsm of particle physics tensor-pseudotensort-ptinteraction is not allowed. Combining the experimental results of an EDM measurement with a closed-shell atom with theoretical values, an estimate of coupling constants can be made. This is a signature of physics beyond the standard model. We have studied the contribution from the tensor-pseudotensort-pt electron-nucleus interaction and Schiff moment using different many-body theories for atomic Yb. I. INTRODUCTION An atom being a composite system, there are many physical phenomena which can lead to an intrinsic atomic EDM. It can arise due to the following: 1. The sub-atomic particles have intrinsic EDM. That is either electrons or protons or neutrons have an intrinsic EDM. 2. The interaction between the sub-atomic particles violates P and T symmetries. The possible interactions are electron-electron, electron-nucleon and nucleon-nucleon. The first case is due to the intrinsic property of the sub-atomic particles. In the current-current interaction formalism one form of interaction that violates P and T simultaneously is the electron-nucleus T-PT interaction [1]. An important feature is that this form of interaction is not allowed in the Standard ModelSM of particle physics. The interaction Hamiltonian corresponding to the electron-nucleus T-PT current-current interaction is H PTV = ic T G F ψn σ µν ψ n ψe γ 5 σ µν ψ e, 2 where C T is the tensor-pseudo-tensor coupling constant, G F is the Fermi coupling constant, ψ n and ψ e are the nucleon and electron fields, γ 5 is the Dirac matrix and σ µν is an antisymmetric tensor got from the Dirac matrices. Within the SM of particle physics the value of the coupling constant C T is zero. If in an atomic system a finite intrinsic EDM is observed and it can be attributed to the T-PT electron-nucleus interaction, then it is a clear indication of nonzero C T, which is a signature of physics beyond the SM. Using the definition of σ µν and treating the nuclear part non-relativistically, H PTV can be simplified to H PTV = i2 2 C T G F I β α ρ N r 1.1 where I is the nuclear spin, α is the Dirac matrix and ρ N r is the nuclear density at r. The presence of the nuclear density implies that H PTV is effective within the nuclear region alone. The P-T-violating property of H PTV can be checked by applying the P and T transformations separately. The effective Hamiltonian H PTV is odd under parity transformation as α transforms like a vector and using Heisenberg equation of motion it can be shown that in relativistic formalism it represents the velocity. Similarly, since the T-reversal transformation has a complex conjugation part it is also odd under T-reversal transformation. The motivation for studying this form of interaction in an atom is that there are non-standard models in particle physics which allow this form of electron-nucleus interaction. An estimate of C T can be made when the theoretical atomic computation is combined with the experimental results and the nuclear structure computation results. Based on 1

2 this, constraints can be put on the validity of the possible non-standard particle physics models. Another important feature of this study is the close connection with the experiments, where each complements the other. To study the atomic EDM arising from the electron-nucleus T-PT-interaction, a closed-shell atom is the right choice as it is the dominating contribution to the atomic EDM. This avoids the direct contribution from the electron EDM, which dominates in the case of open-shell atoms but the electron EDM still can contribute through the hyperfine interaction. Here atomic Yb has been chosen as it is a closed-shell atom and has high Z. In addition, it has interesting many-body effects as it is a rare earth element. II. THE LOWEST ORDER ATOMIC EDM In this paper the orbitals with the total angular momentum j = l 1/2 are denoted by *, that is p and p represent p 1/2 and p 3/2 respectively. To compute the intrinsic EDM of an atom arising from the electron-nucleus T- PT-interaction, the sp approximation will be used. In this approximation only the matrix elements between orbitals of s and p symmetries are computed. This follows from the physical condition that only these orbitals are non-zero within the nuclear region. The Dirac-Coulomb atomic Hamiltonian is H atom = c α i p i + β 1c 2 V nucr i r i i 2 r j ij = i t i + U DF r i U DF = H DF + V es, r ij ij where V nuc r i is the nuclear potential experienced by the i th electron, U DF is the Dirac-Fock central potential, H DF is the Dirac-Fock Hamiltonian and V es is the residual coulomb interaction. From here on, atomic EDM will be used instead of referring explicitly as the intrinsic EDM of an atom. When computing the matrix elements, the relativistic two component orbital notation is used. In this notation an orbital ψ is an eigenfunction of the Dirac-Fock Hamiltonian. It is identified by the principal quantum number n, angular momentum quantum numbers κ and m; in spherical polar coordinates it is represented as ψ = 1 r Pnκ r χ κm θ, φ iq nκ r χ κm θ, φ, 2.2 where P r is the large-component, Qr is the small component, χ κm θ, φ and χ κm θ, φ are the corresponding angular parts. The atomic orbitals ns and n p can also be expressed in the same form. After the angular integration, the matrix element of H PTV between ns and n p orbitals takes the form n nsh PTV p = 2 C T G F σ Nz 0 dr P n 1 rq n 1r 1 3 Q n 1rP n 1r ρ N r. 2.3 That is, the angular integrations introduce constant multiplicative factors. A similar expression for other matrix elements can also be derived. Since H PTV is anti-hermitian, from 2.3 its matrix elements are real and hence n n p H PTV ns = nsh PTV p. 2.4 Although the radial integration has limits from 0 to, it is required only within the nuclear region as ρ N r beyond it is zero. When computing the matrix elements, it is important to get an accurate radial part of the orbitals and the nuclear density. A fairly accurate model of the nuclear density which agrees quite well with the experimental results is the Fermi-nucleus. In this model the nuclear density is given by ρ N r = ρ exp r b a where ρ 0 is a constant, b is the half density radius as ρ N r = ρ 0 /2 and a is related to the skin thickness t as t/a = 4 ln 3., 2

3 Atomic Yb is a closed-shell atom with 6s as the outermost occupied shell and the virtual p orbitals are n > 5p. Introducing H PTV as a perturbation to the atomic Hamiltonian produces admixture to 6s from orbitals of other symmetries opposite in parity. As mentioned earlier, among the virtual orbitals only the n p orbitals contribute to the admixture. The new orbitals are no longer parity eigen-states as they are of mixed parity. Representing the mixed parity orbital by 6s, using perturbation theory this can be written as 6s = 6s + 6s 1 n 6s p H = 6s + n PTV p, ɛ 6s ɛ n p where 6s 1 is the mixing from the opposite parity orbitals due to H PTV, ɛ 6s and ɛ n p are the orbital energies of 6s and n p respectively. This is the lowest order effect without any many-body effects from the residual coulomb interaction. Computation at the many-body level can be done in many ways depending on the many-body effects included, and a few methods will be used and discussed in later sections. The atomic EDM D a is the expectation value of the dipole operator D = r with respect to the mixed parity orbital 6s [2] D a = 6sD 6s 6s Dn p n 6s p H PTV = 2, 2.5 ɛ 6s ɛ n p n =6 where the multiplication factor 2 is to include the complex conjugate term which is identical to the normal term due to the following: first, the dipole operator is hermitian, diagonal and its matrix elements are real hence ns D n p = n p D ns and second, as mentioned before, n p H PTV ns = ns H PTV n p. Another method of derivation which gives the same result is the method of linear response to an external electric field. This formulation is more appealing as it has direct bearing on experiments to detect the atomic EDM. From the above expression it is clear that the atomic EDM is directly proportional to the matrix elements of H PTV and D, and inversely proportional to the energy difference ɛ 6s ɛ np. This gives a criterion for choosing the an atom for experiment; Yb is a good choice as the energy denominator, ɛ 6s ɛ 6p, of the largest contribution to lowest order is very small. n =6 III. ATOMIC EDM WITH MANY-BODY WAVE-FUNCTIONS A. Atomic EDM with CI Wave-Functions Let { ψ i } be a set of orbitals which are eigen states of the Dirac-Fock Hamiltonian or other single-electron Hamiltonian with an apropriate central field potential. Using these orbitals, a set of configuration state functionscsfs { Φ i } can be constructed. A CSF is a linear combination of determinants and is identified by the quantum numbers γ, total angular momentum J and total magnetic quantum number M, where γ is an additional quantum number required to define each CSF uniquely. The atomic state functionasf can then be obtained as a linear combination of these CSFs. An ASF is defined by the same J and M but with an additional quantum number Γ to identify each of the ASF uniquely ΨΓ i JM = j Φγj C ij JM. The ASFs are eigen-functions of the Dirac-Coulomb atomic Hamiltonian H atom and satisfy the equation ΨΓi ΨΓi H atom JM = E i JM, where E i is the energy eigenvalue of the ASF. When computing matrix elements of operators, a sum over M is performed and effectively it is the quantum numbers Γ and J that identify an ASF. In addition, as H atom commutes with the parity operator P, the CSFs and ASFs are parity eigenstates. In the CI method the CSF co-efficients C ij are obtained after diagonalizing the atomic Hamiltonian within the CSF space [9]. Let { Φ i γ i JM} and { Φ i γ i J M } be CSF spaces of opposite parities. Diagonalizing the atomic Hamiltonian within these CSF spaces gives two sets of ASFs { Ψ i Γ i JM} and { Ψ i Γ i J M }, which are opposite in parity. In general Ψ i represents an ASF in { Ψ i Γ i JM} with quantum numbers Γ i, J and M, similarly Ψ i represents the corresponding ASF in opposite parity space with quantum numbers Γ i, J and M 3

4 Let Ψ 0 { Ψ i Γ i JM} be the ground state ASF. The PT-violating interactions within the atom introduce opposite parity corrections to the ASFs. Let Ψ 0 be the mixed parity ground state, and using perturbation theory it can be written in terms of the CI wavefunctions as Ψ 0 = Ψ 0 + I Ψ I Ψ I HPTV Ψ0 E 0 E I, where Ψ I { ΨΓJM}, E I and E 0 are the energies of the opposite parity ASFs and the ground state ASF. The atomic EDM is then the expectation value of the dipole operator with respect to Ψ 0 : D a = Ψ0 D Ψ0 = 2 I Ψ 0 D ΨI Ψ I HPTV Ψ0 E 0 E I, which in terms of CSFs can be written as D a = 2 Iijkl C 0i C Ij C Ik C 0l Φ i D Φj Φ k HPTV Φl E 0 E I. This is the required expression of the atomic EDM in terms of the CSFs. This approach requires two diagonalizations, one each in the two opposite-parity CSF subspaces. When the number of the CSFs in these subspaces is large, the diagonalization approach is less desirable in terms of computational efficiency. B. Atomic EDM with Perturbed CI Method The groundstate CSF Φ 0 by itself does not have any dynamic electron-correlation effects but an ASF Ψ 0 obtained after a CI computation includes all the electron-correlation effects within { Φ i }. Introducing the parity and timereversalpt violating interaction Hamiltonian H PTV as a perturbation, the total atomic Hamiltonian assumes the form H = H atom + H PTV. As H PTV does not commute with the parity operator, the eigenfunctions of H are no longer parity eigenstates. It introduces an opposite parity correction to the wave-function and since H PTV scales as G F it should be included to first order only. The Schrödinger equation assumes the form Ψ0 H Ψ 0 = E 0 Ψ0 = E 0 +, where Ψ 1 0 is the mixing to the ground state ψ 0 from the opposite parity atomic states due to H PTV. The perturbation H PTV introduces no energy correction as it is odd in parity. Expressing Ψ corr 0 as a linear combination of opposite parity configurations, the ground state ASF of H is Ψ 0 = Ψ 0 + Ψ 1 0 = Φi C 0i + Φm C 0m, 3.1 i m Ψ 1 0 where C 0m are the correction coefficients first order in H PTV but all order in V es within { Ψ} and { Ψ}. Introducing the perturbation parameter λ, the Schrödinger equation becomes H atom + λh PTV i C 0i Φi + λ m C 0m Φm = E 0 i C 0i Φi + λ m C 0m Φm. 3.2 Now project the equation onto Ψ corr 0 and retain the first order terms in λ. This gives the matrix equation: E 0 H 0 C = H + C. PTV 3.3 4

5 Solving this matrix equation will give the required coefficients of the opposite parity CSFs C 0m. The expectation value of the atomic EDM can then be computed using the perturbed ASFs as D a = Ψ0 D Ψ0 = 2 C 0i C 0j Φ i D Φj, ij where D is the dipole operator. Unlike the earlier approach, in this approach only the { Φ i γ i JM} CSF space needs to be diagonalized. The set of linear equations can be solved very efficiently using algorithms like conjugate gradient method. In atomic Yb the filled 4f shell is very shallow and the 5d is unfilled, as a result excitation from 4f and to 5d shells are important to include the many-body effects correctly. These shells being of high j, unfilled shells of these orbitals produce a large number configurations. Although the diagonalization has been reduced to only one of the configuration subspace, it is undesirable when the number of configurations is large, which is true in systems like atomic Yb. The diagonalization can be avoided altogether by using Bloch equation based Many-Body methods. C. Atomic EDM using MBPT Many-Body Perturbation TheoryMBPT at the single particle level was used in atomic structure calculation by Kelly [10 12] for the first time to compute correlation energies. Here the computations are at the level of configurations. The V es part in equation2.1 can be treated perturbatively by partitioning the configuration space into model and complementary spaces { Φ i P } and { Φ i Q } respectively [7]. The model space has configurations which mix strongly with the ground state configuration and remaining the configurations are included in the complementary space. The projection operators for these CSF spaces are, Φ i Φ P P i Q = Φ i Φ Q Q i and P + Q = 1. P = i i The mixing from the complementary space is computed using Rayleigh-Schrödinger perturbation theory. The essence of Rayleigh-Schrodinger perturbation theory is the Bloch equation [13] [Ω es, H 0 ] P = QV es Ω es P χ es P V es Ω es P, 3.4 where χ es = n=1 Ωn es and can be written in recursive form as [ Ω n es, H 0 ] P = QV es Ω n 1 es n 1 P Ω m es P V es Ω n m 1 es P. 3.5 To compute the atomic EDM, one more perturbation H PTV or H dip is to be introduced. Considering H PTV as a perturbation, the total atomic Hamiltonian assumes the form m=1 H = H 0 + V es + λh PTV, where λ is the perturbation parameter. These perturbations introduce corrections to the configurations in P -space; corrections from V es are of same parity as the unperturbed CSF but the corrections from H PTV are opposite in parity. The residual coulomb interaction V es is to be treated to all order but H PTV should be treated to first order only. The exact Schrödinger equation is H Ψ = E 0 Ψ, 3.6 where Ψ = Φ 0 P + j C j Φj Q + k Φk Φ0 C k Q = Ω es + λω es,edm P, where Ω es,edm is the wave-operator all order in V es but one order of H PTV. The wave operators Ω es,edm and Ω es are computed from the modified Bloch equation 5

6 ] n 1 [Ω n, H 0 P = QH Ω n 1 P Ω m P H Ω n m 1 P, 3.7 where H = V es + λh PTV. Define Ω n edm as the n th order wave-operator required for the EDM computation. This can be obtained from 3.7 as Ω n edm = Ω n es + λ Ω n = Ω n es + λω n 1 es,edm λ. 3.8 λ=0 The evaluation of first order differential with respect to λ at λ = 0 is to select out the linear terms in λ, and these are the terms which has one order of H PTV. The total wave-operator Ωedm is defined by the total order of the perturbations but while writing out the components Ω es and Ω es,edm the superscript denote the order of residual coulomb interaction and the additional subscript edm denote the presence of H PTV as perturbation. The first order wave-operator is ] [Ω 1, H 0 P = Q V es λh PTV P. Using the resolvent operator, R = Φ i Φ Q Q i i E 0 H 0, gives: Ω 1 P = R V es + λh PTV P. 3.9 As the two perturbations are opposite in parity, it is appropriate that the model space and the complementary space be separated into subspaces of opposite parities. Then the model space projection operator can be divided into P + and P. Doing the same with complementary space projection operator and the resolvent operator we can write With these definitions 3.9 can be rewritten as Ω 1 P = m=1 P = P + + P ; Q = Q + + Q ; R = R + + R. R + V es + λr H PTV P + + R V es + λr + H PTV P To make the derivation less cumbersome, assume that the model space has configurations of only one parity i.e. P = P +, then Ω 1 P = R + V es + λr H PTV P + = Ω 1 es + λω 0 es,edm P +. The expression for the wave-operator has been split according to the definition given earlier, according to which indicing of wave-operator Ω es,edm is based on the order of V es, e.g. Ω 0 es,edm has one order of H PTV but no V es at all. Similarly the expression for the second order wave-operator is ] [Ω 2, H 0 P + = Q V es + λh PTV Ω 1 P + QΩ 1 P + V es + λh PTV P +. Using the definition of 3.8 Ω 2 edmp + = R + V es R + V es R + R + V es P + V es P + + λ R V rmes R H PTV + R H PTV R + V es R R H PTV P + V es P +, = Ω 2 es + λω 1 es,edm P The other higher order wave-operators can be evaluated in the same way. In general the wave-operator Ω n es,edm can be obtained from the modified Bloch equation [ ] Ω n es,edm, H 0 P + = Q H PTV Ω n es + Q V es Ω n 1 6 es,edm n 1 m=0 Ω m es,edm P +V es Ω n m 1 es P +.

7 In writing the above equation, Ω 1 es,edm = 0 has been used and the renormalization term start contributing from n=1 onwards. The above equation is valid starting from n=0. The value of the atomic EDM is then Ψ D Ψ D a = Ψ. Ψ Because D is an odd parity operator connecting CSFs of opposite parities, only those terms linear in λ need to be retained, and we get Ω D a = Φ 0 esdωes,edm Ψ0 Ω + Ψ Φ0 0 es,edmdω es = Φ Da 0 Φ 0, 3.12 eff where D a eff = P Ω es DΩ es,edm +Ω es,edm DΩ es P is the effective atomic EDM operator and is different from the usual effective dipole operator. The usual effective dipole operator D eff used in computing dipole transition amplitude connects two CSFs of different parities. The effective intrinsic atomic EDM operator is an expectation value operator and the wave-operators in it has H PTV and residual coulomb operators as perturbations where as in D eff it is only H es. IV. COMPUTATION OF ATOMIC EDM A. The Orbitals and the Configurations The orbital basis set used in the CSF based perturbation theory can be of any form but it should satisfy the following properties: 1. completeness 2. orthonormality. These conditions are satisfied by a set of orbitals generated using a single particle Hamiltonian like the Dirac-Fock potential. Similarly, a set of orbitals generated using the [16] V N 1 potential satisfies these conditions, too. We have used V N 1 orbitals in our calculation. The completeness criterion of the orbital space is determined by the convergence of the property of interest and the ground state energy E 0. For atomic Yb the occupied-orbitals are generated first with the ground state configuration 6s 2. The virtual orbital ψ i is generated using the configuration 6sψ i where the orbitals till 6s are frozen. The total angular momentum of the configuration is chosen as the smaller of the two obtained after coupling the angular momentum of 6s and ψ i. The disadvantage of using the V N 1 potential is that to make the orbital space complete, the positive continuum spectrum must be included. Though the continuum contribution to EDM is not very significant, the quadrature of the matrix elements involving continuum orbitals always incur errors. Energies of the occupied and the bound virtuals are tabulated in Table I according to symmetry. In order to check on the completeness of the bound virtual orbitals E 0 is computed with the addition of each orbital of different symmetries. For each symmetry the orbital space is increased by one at a time and E 0 is computed using CI. The CSF space for CI is the ground state CSF Φ 0 and the set of CSFs which has same angular momentum and parity as Φ 0 generated using the virtual orbitals of the symmetry. Completeness of the virtual orbital space is assumed when E 0 converges to the fourth place of decimal. Ground state energy for each of the symmetry are given in the Table II. In the Table II the change in energy is the difference between E 0 of the present set of CSFs and the earlier set. For each symmetry the starting comparison is the ground state CSF energy. Table II shows that the convergence pattern is different for each symmetry. Compared to 7s, the inclusion of 6p orbital introduces a larger change in E 0 and is a result of 6p being closer in energy to 6s than 7s. But the change in E 0 with the inclusion of 7p is much smaller than with the inclusion of 7p. Since the energy of 7p is higher than 7p, the energy separation from 6s cannot explain this. This is due to the configuration mixing between those other than Φ 0. Energy values for the f and f are not included as they converge with a single virtual orbital. For the computation, six bound virtual orbitals for each symmetry is taken, the orbital space is 1-12s, 2-11p, 2-11p, 3-10d, 3-10d, 4-10f and 4-10f. In the continuum orbital space the number of points used in the Gauss-Laguerre quadrature decides the number of orbitals to be included [17 19]. The continuum orbitals are identified by the symmetry and the linear momentum 7

8 k, where 0 k. They are assigned negative principal quantum numbers to distinguish from the bound orbitals and are generated by using the energy ɛ k = k 2 /2. To include the complete continuum spectrum the contribution from the continuum orbitals has to be integrated over the whole spectrum. Consider the lowest order contribution to the atomic EDM with the continuum orbitals as the intermediate orbital EDM cntm = 0 dk ψ 6s D ψp 1 ψ6s k ψ P k H PTV Ek = 0 dkfk, where Ek = ɛ 6s ɛ P k and fk = ψ 6s D ψ P k ψ P k H PTV ψ 6s Ek 1. Quadrature within the continuum spectrum can be simplified by discretizing it using the Gauss-Laguerre quadrature. This reduces the integration over the entire continuum spectrum to a sum over the roots of the Laguerre polynomials. If n is the number of roots used in the Gauss-Laguerre quadrature and the k i the i th Laguerre root the above integration reduces to EDM cntm = n w i e k i fk i. i=1 With this form of quadrature, the continuum orbitals are required only for energies corresponding to the roots of the Laguerre polynomials. For convenience the roots are normalized to 20 such that the new roots are k i = 20 k i/k n and the corresponding energies are given by 2 20 ki. ɛ ki = 1 2 k n Using the normalized roots the integration is 20 n EDM cntm = w i e k i fk i. 4.1 k n This is the required integration for the continuum spectrum [20]. The same expression can also be used for CSFs by reducing the matrix elements to the orbital level. i=1 B. Wave-Operator Computed in terms of order of Perturbation The derivations in the previous section is general and the atomic Hamiltonian can be partitioned into any convenient form. For computation Epstein-NesbetEN partitioning [14] will be used. In the following derivations the indices i and j cover all the CSFs within the configuration space considered and H PTV is to be treated as perturbation. With EN-partitioning the unperturbed Hamiltonian is defined as H 0 = HatomΦi Φi HatomΦi Φi. Φ i Φ i + Φ i Φ i 4.2 i To distinguish the residual coulomb interaction from V es defined earlier, it will be denoted by H es and connects different CSFs. Similar to H 0 it can be defined as operator in terms of CSFs as H es = HatomΦj Φi HatomΦj Φi. Φ i Φ j + Φ i Φ j 4.3 i,j,i j In EN-partitioning the effect of diagonal ladder diagrams is taken to all orders in first order wave-function and second order for energy. The energy of CSF Φ i in EN-partitioning is the expectation value of H atom and the Schrödinger equation for the unperturbed atomic Hamiltonian H 0 is Φi Φi HatomΦi H 0 = E i where E i = Φ i. i i,j,i j The definitions of P and Q do not change, but the energy denominator changes. Similarly, H PTV assumes the form 8

9 H PTV = i,j Φ i Φ i HPTV Φj Φ j = i,j Φ i HPTV Φj Φi Φ j. 4.4 That is, the unperturbed Hamiltonian and the perturbations are cast in operator form within the configuration space considered. The interaction Hamiltonian H PTV map from { Ψ} to { Ψ} and H PTV can connect from { Ψ} to { Ψ} only when H PTV is treated to second order but as H PTV is to be treated to only first order. Consider the expression for Ω 1 es, in terms of these definitions it can be written as Ω 1 es P + = i,j Φ j Q Q Φ j Hatom Φi E i E j P P Φ i = i,j Q Φ j Hatom Φi E i E j P Φj Q P, Φ i where i P, j Q and E i = Φ i H atom Φ i, the term within parentheses in the expression on the right hand side is just a number. The expression can be rewritten as Ω 1 es P + = HatomΦi C 1 ji es Φj Φ 1 Q P Φ i where C ji es = Q j P. E i,j i E j The second order wave-operator in terms of matrix elements is Ω 2 es P + = Φ j Φ Q Q j Φi H es Ω 1 es Ω 1 Hes es Φ k Φ P P k E i,j i E j where k P +. Substituting the expressions for H es and Ω 1 es [ HatomΦl Φ j Ω 2 es P + = i,j l Q QC 1 li es + k k P P, Φ i in terms of the matrix elements we get ] Φj C 1 jk es HatomΦi Φ Q P i P Φ k P E i E j, where l Q, the whole expression within the square brackets is just a number, the wave-operator second order in residual coulomb interaction can be written as Ω 2 es = C 2 ji es Φj. Φ Q P i ij This can be extended to higher orders and summing all orders the full wave-operator can be defined as Ω es = C n. ji Φ j Φ Q P i 4.5 nij A similar expression for the wave-operator Ω n es,edm can also be obtained. Consider the expression for Ω0 of the operators defined earlier it can be written as Ω 0 es,edm P + = HPTVΦi Φ j Φ Q Q j Φ PP i = HPTVΦi Φ Q j P Φ j Q P Φ i E i,j i E j E i,j i E j = C 0 ji es, edm Φj. Q P Φ i i,j es,edm, in terms Like in the case of Ω es, the above expression can be generalized to any order n by using the Bloch equation and summed to get the full wave-operator Ω es,edm = C n. ji es, edm Φ j Q P Φ i 4.6 nij Using the expression for Ω es and Ω es,edm the wave-operators Ω es and Ω es,edm can also be derived. represented as Ω n es = C n, ji es Φ j P Q Φ i Ω n es, edm = C n. ji es, edm Φ j P Q Φ i ij ij Let these be The wave-operators Ω es and Ω es,edm have the following properties: 9

10 1. The co-efficients in Ω es and Ω es,edm are real as these are product of real matrix elements, hence C n ji es = C n ij es and C n ji es, edm = C n es, edm. ij 2. These wave-operators are non-hermitian ie Ω es Ω es and Ω es,edm Ω es,edm. This is evident from the form of the projection operator part in the expression of the wave-operators. 3. The wave-operators are state specific, Ω es and Ω es,edm can act only on the bra CSF Φ 0 and not on any other ket nor bra. Similarly, Ω es and Ω es,edm can act only on the ket CSF Φ 0. The atomic EDM in terms of the wave-operators can be computed using the expression V. RESULTS A. The Lowest Order EDM The expression for the lowest order contribution to EDM from the virtual orbital np is Da np = 6s D np np H PTV 6s ɛ 6s ɛ np. where ɛ 6s and ɛ np are the single particle energies of 6s and np orbitals respectively. Combining both H PTV and dipole terms the lowest order contributions to EDM from the first few virtual p orbitals are tabulated in Table III. The values in this table are parameterized in terms of C T σ Nz ea 0. This is at the level of single particle. A similar computation can be done at the level of configurations using CSFs instead of single particle orbitals. The expression for the EDM in terms of the CSFs is 6s 2 J=0 6s D6snp J=1 6snp J=1 H 2 PTV J=0 Da =. E 0 E np np Where E np is the energy of the CSF 6snp J =1. The values got using the single particle approach and the CSFs should be the same except for the effect of the static correlation, which will change the value of the energy denominator from the single particle energy denominator. From Table III it is clear that at the single particle level the lowest order contribution is dominated by 6p and the contribution from the other virtual p orbitals is marginal. Like in 7s there is a flip in the sign of H PTV matrix element for the intermediate energy virtual p orbitals but a corresponding sign flip in the dipole matrix element maintains the sign of EDM. If not for the accompanying sign flip it can lead to cancellations. The fall in the contribution from the virtual p starting from n=7 is due to a decrease in the H PTV coupling and the widening energy gap between np and 6s. Compared to 6p the D coupling with 7p is more than one order of magnitude down and the energy gap is almost twice. In addition H PTV coupling also decreases but at a slower pace compared to D coupling. Over all there is a rapid fall in the D coupling till 9p after which it continues to fall but at a slower pace. The magnitude of the energy difference falls very fast for the first two bound virtual p but is relatively stable after as the energy separation become smaller for the high lying orbitals. The contribution from the continuum orbitals to the lowest order can also be computed in the same way. To include the whole of continuum p orbitals space first we have to compute the required matrix elements and apply the Gauss-Laguerre quadrature. The individual contributions from the continuum p orbitals are as in Table IV. The last column in the table gives the total value of EDM till that continuum orbital. Though the 12 point Gauss-Laguerre quadrature has been used only the first 9 points have been given as the contribution from the rest is very small. The quadrature can be done using the expression 4.1. Figure 1 shows the EDM contributions from the bound and continuum contributions plotted against the orbital energies. For the continuum orbitals, the plot is generated with more data points than required for the Gauss-Laguerre quadrature. Trends in the contribution from the bound and continuum orbitals are very different and this is clearly brought out in the plots. The column for D a in Table IV is the individual contribution from the continuum orbitals weighted by the corresponding Laguerre weight factors. The last column is the accumulated value of D and the final value is the total contribution from the continuum orbitals. The individual contribution from 1p is larger than the contribution from 7p but its contribution to the overall quadrature gets suppressed as the weight factor is less than unity. The dipole matrix element flips in sign with increase 10

11 in energy but magnitude of the energy denominator and the H PTV matrix element increase monotonically. This also flips the sign of the contribution to EDM. Since the energy has been scaled to 20, the contribution from the whole of the continuum p spectrum is the final value multiplied by 20/k 12, which gives the contribution from p continuum orbitals to the lowest order EDM as C T σ Nz ea 0. Adding up with the contribution from the bound np virtuals, the value of EDM is C T σ Nz ea 0. An important feature of the continuum is that the contribution from 1p is quite large compared to the contribution from the whole of the continuum spectrum. This implies that the important contribution from the continuum is from a small region in the k-space near the origin. This is evident from the individual values in Table IV. For completeness of the computation, the continuum orbitals is important as the contribution from the continuum is larger than that of 8p. At the single particle level 6p is the most important virtual orbital, it accounts for % of the lowest order EDM. The remaining p orbitals including the continuum accounts for only 2.225% and the continuum alone contributes only 0.296% to the total value. B. Computations with Many-Body Wave-Functions As described in the previous chapter, atomic EDM can be computed by diagonalizing within the whole configuration space. It involves two diagonalization, one within each of the even and the odd-parity sub-spaces. Since atomic EDM is the expectation value of the dipole operator with respect to the ground state, for atomic Yb only the lowest energy ASF in the even parity sub-space is required. Within the odd-parity sub-space, all the ASFs are required as they are the intermediate states. Though the 5d orbital does not contribute to the atomic EDM directly, it is very important for the correlation effects as there are CSFs containing 5d which mix strongly with Φ 0. Other configurations that mix strongly with Φ 0 are the double excitations from 6s 2 to the lower s and p orbitals. Among the odd-parity ASFs the most important CSF is 6s6p and similar to the even-parity sub-space configurations with excitations to 5d are important. For comparison, the value of EDM computed with CI, the perturbed CIPCI and the Bloch equation based formulation are as tabulated in Table V. The unit for EDM used in Table V is the same as in earlier tables. The CSFs considered are single excitations from 6s,4f and 4f and double excitations with the occupied-orbital configuration 4f 13 6s 1. The sequence of the CSFs is arranged in increasing principal quantum numbers and corresponds roughly to increasing energy. The results using PCI have been given for both the interactions treated as perturbations separately. Results under the heading H PTV are computed treating H PTV as the perturbation in the PCI formulation and similarly for D. CI and PCI are diagonalization based methods and include the residual coulomb interaction within the configuration space to all order, whereas in the Bloch equation based MBPT approach, the order of residual coulomb interaction is decided by the convergence criterion. To check the convergence, the absolute value of the change in the CSF coefficients for two consecutive orders are computed. The wave operator is considered converged when the maximum value of this change in CSF coefficients is less than the convergence criterion. For the results tabulated the wave-operator is computed until the order which contributes more than to the CSF coefficients with the residual coulomb interaction. With the H PTV included as perturbation two sets of computation are done; one with the convergence criteria set to 10 4 and another to These are tabulated in the columns labeled as [ 4] and [ 8] respectively. Higher convergence criteria with H PTV has been chosen as it has a scaling factor of G F. Results in the table emphasizes the following points: 1. The results from CI and PCI are in perfect agreement thereby suggesting that the physical effects incorporated in the two approaches are the same even though the formulations are different. The value of EDM computed with PCI is the same whether H PTV or D is treated as the perturbation. 2. The result does not change significantly with the inclusion of CSFs which has excitations to high lying orbitals. This is another test for the convergence of the orbital basis set in terms of all the interactions involved in the computation where as the test for convergence in the single particle picture was without the residual coloumb interaction. Although the details of the formalisms are different, the physical effects included in all the procedures are the same, so the results from MBPT should be identical with the CI and PCI results. Considering the first column in the MBPT results where the convergence criterion is set to 10 4, this result differs from those of CI and PCI by an order of This can be accounted as due to different physical effects included. A higher value of the convergence criterion mean lower order of residual Coulomb interaction in the odd-parity configuration sub-space. As shown in the Table V when the convergence criteria is decreased to 10 8, there is almost perfect agreement between the results from various methods. When the convergence criteria is set to 10 4, the order at which the wave-operator converges is 22 for Ω es 11

12 in the even-parity CSF space and 12 for Ω es,edm in the odd-parity space, which means 22 and 12 orders of residual Coulomb interaction is included in the even-parity and odd-parity configuration spaces respectively. This establishes the one-to-one correspondence of the physical effects included in each of the methods. The MBPT based approach has the best efficiency among the various approaches in terms of the memory usage and execution time. Considering the last case with 500 even parity CSFs and 1500 odd-parity CSFs, the time taken with CI, PCI and MBPT approaches are 5:58:23,00:15:39 and 00:7:07 hrs, respectively. The ratio of the time taken with execution time of CI normalized to 100 is 100:4.366: This clearly shows the advantage of the MBPT based formalism in terms of execution time. For the MBPT approach, the time taken is with the convergence criterion set to The part of the CI and PCI approaches which takes most time is the matrix diagonalization and the enhanced run time efficiency of MBPT is the absence of diagonalization. As shown in the Fig. 2, the convergence of Ω es is much faster than that of Ω es,edm. This is due to the strong coupling with H PTV. However one point to be noted is that H PTV is parameterized in terms of G F, which will be included later while computing D a. The computation has been done with excitations from 6s,4f and 4f alone. This checks the importance of the high-lying bound virtuals. It is also necessary to check the importance of excitations from the occupied orbitals, for which only the lowest virtuals in each symmetry are selected and the configurations with excitations from deeper cores are included gradually. The result for such a sequence of runs is tabulated in Table VI. The sequence of runs show that the excitations from the 4f orbitals contribute significantly to the value of EDM but the excitations from the 5p and 5s are not so important. This justifies the choice of excitations till the 5s occupied orbital. Configurations with excitation from the 4f occupied orbital adds to the contribution from the configurations with excitations from 6s but there is a cancellation with the contributions from the 5p and 5s excited configurations. VI. CONCLUSION The use of the V N 1 nessecitates the inclusion of the positive continuum orbitals to satisfy the completeness condition but their overall contribution to the atomic EDM is very small yet for a high precision computations their inclusion is desirable. At the lowest order the contribution from the contribution from the continuum is a mere 0.296% to the final value of C T σ N ea 0. The computation with the many-body wave-functions indicates an enhancement in the atomic EDM, which implies the importance of many-body effects. This is due to the unfilled 5d shell and the loosely bound 4f shell. The final atomic Yb EDM is the contribution from the T-PT electron-nucleus interaction and the Schiff moment. To estimate the contribution from the Schiff moment, results from nuclear structure computations are required. The T-PT electron-nucleus is dependent only on the nuclear spin whereas the Schiff moment is dependent on the nuclear moment. If these different dependence can be exploited in an experiment, then it should be possible to separate out these two contributions. We have also done the calculation of the contribution to atomic Yb EDM from the Schiff moment for the electronic component and this shall be the topic of a later publication. The results shows that the CI, PCI and Bloch equation based MBPT with EN-partitioning are identical provided MBPT calculation is done to very high order. A drawback of these methods is that they are size-consistent when the CSF space considered is incomplete. It is difficult to make the CSF space complete due to presence unfilled 5d shell and filled 4f, as number of CSF increases by large amount when these shells are partly filled. In a paper to follow, we will present the remedy to the problem of size-inconsistency and consider a larger CSF space. VII. ACKNOWLEDGMENTS We would like to thank Prof. Debashis Mukherjee and Dr. Rajat Choudhury, for many discussion we had with them. [1] S.M. Barr, Int. J. Mod. Phys. A 8,209,1993. [2] Ann-Marie Martensson-Pendrill, Methods in Computational Chemistry, Volume 5: Atomic and Molecular Properties, Ed by Stephen Wilson, 99Plenum Press, New York, [3] L. I. Schiff, Phys Rev 132, [4] P. V. Coveny and P. G. H. Sandars, J. Phys. B 16,

13 [5] I. B. Khriplovich, Parity Nonconservation in Atomic PhenomenaGordon and Breach, Philadelphia, [6] Harry M. Quiney, Jon K. Laerdahl, Knut Fægri Jr. and Trond Saue, Phys Rev A 57, [7] Ingvar Lindgren and John Morrison, Atomic Many-Body TheorySpringer, New York, [8] J. Goldstone, Proc. Roy. Soc.London A239, [9] Takashi Kagawa, Yoshie Honda and Shuji Kiyokawa, Phys. Rev. A 44, [10] Hugh P. Kelly, Phys. Rev. 131, [11] Hugh P. Kelly, Phys. Rev. 132, , Hugh P. Kelly, Phys. Rev. 134,A [12] Hugh P. Kelly, Phys. Rev., 136,B [13] C. Bloch, Nucl. Phys. 6, [14] Paul. S. Epstein, Phys. Rev. 28, [15] R. K. Nesbet, Pro. Roy. Soc. London, A [16] Hugh P. Kelly Phys. Rev. 131, [17] W. F. Perger and Vasan Karighattam, Comp. Phys. Comm. 66, [18] W. F. Perger, Z. Halabuka and D. Trautmann, Comp. Phys. Comm. 76, [19] Handbook of Mathematical Functions, Edited by M. Abromowitz and A. StegunDover Publications, [20] W. F. Perger, Ph.D. Thesis submitted to Colorado State Univ [21] I. P. Grant, Adv. Phys. 19, [22] V. V. Flambaum, I. B. Khriplovich and O. P. Sushkov, Nuclear Physics, A 449,

14 TABLE I. The single particle orbital energies Orb. Energy Orb. Energy Orb. Energy 1s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f s p d f p p d f p p d f p p d p p d TABLE II. Table II: CI-energy for each of the symmetry with increasing CSFs Sl. No. CSFs Energy Change in Energy 1 6s 2 + 7s 2 + 6s7s s s8s s s9s s s10s s s11s s s12s s 2 + 6p p 2 + 6p 7p p p 8p p p 9p p p 10p p p 11p s 2 + 6p p 2 + 6p7p p p8p p p9p p p10p p p11p s 2 + 5d d 2 + 5d 6d d d 7d d d 8d d d 9d d d 10d s 2 + 5d d 2 + 5d6d d d7d d d8d d d9d d d10d

15 TABLE III. Lowest order contribution to EDM at the single particle level. n 6s D np np H PTV 6s ɛ 6s ɛ np D a np Cumulated D a np TABLE IV. Lowest order contribution to EDM from continuum orbitals. n 6s D np np H PTV 6s ɛ 6s ɛ np Da Cumulated D a TABLE V. Comparison of results from different methods. Sl. No No of CSFs CI PCI MBPT Odd Even H PTV Da [ 4] [ 8]

16 TABLE VI. Contributions from the occupied orbitals. Sl. no CSFsnon-rel Value of D a Even CSFs Odd CSFs 1 6s 2 + 7s 2 + 6p 2 + 5d 2 6s6p + 7s6p + 6p5d f 2 + 6s7s + 6p5f + 5d5f + 4f 13 6p + 5f + 4f 13 7s + 5d + 4f 13 6s7s6p + 7s5f + 4f 13 6s7s 2 + 5d 2 + 7s5d 2 +6p5d + 5d5f + 4f 12 7s 2 +6p5f + 4f 12 7s6p p 2 + 5d 2 + 5f 2 + 7s5d +7s5f + 6p5d + 5d5f +6p5f + 5p 5 6p + 5p 5 6s7s6p + 5p 5 7s + 5d + 5p 5 6s7s 2 +7s5f + 6p5d + 5d5f +5d 2 + 7s5d + 6p5f 3 + 5p 5 4f 13 7s 2 + 6p 2 + 5d 2 + 5p 5 4f 13 7s6p + 7s5f + 6p5d f 2 + 7s5d + 6p5f +5d5f + 5p 4 7s6p + 7s5f + 5p 4 7s 2 + 6p 2 + 5d 2 + 5f 2 +6p5d + 5d5f +7s5d + 6p5f + 5s7s + 5s6s7s 2 + 5f 2 + 5s6p + 5s6s7s6p + 7s5d +5d 2 + 5f 2 + 7s5d + 6p5f +6p5d + 5d5f + 5s4f 13 7s s4f 13 7s6p + 7s5f + 6p5d+ +6p 2 + 5d 2 + 5f 2 + 7s5d d5f + 5s5p 5 7s6p +6p5f + 5s5p 5 7s p 2 +7s5f + 6p5d + +5d5f +5d 2 + 5f 2 + 7s5d + 6p5f 16

17 Energy of the np* orbitals The energy of the continuum orbitals. a b FIG. 1. The contribution from the continuum orbitals a b FIG. 2. The convergence of the wave-operators 17