# Presentation: 546619

## The Use of Virial theorem and Sum rule in Atomic Structure Calculation

**Department of Physics**
**University of Nevada Reno, **
**Reno, NV 89557,USA**

**The Use of Virial theorem and Sum rule in Atomic Structure Calculation**

## Motivation:

-  Importance of two-electron systems
-  	-easy to interpret spectra
- 	-useful in the diagnostics of astrophysical plasmas
-  How can we make sure that we have globally the best
-     solution.
-  For bound state calculations the standard virial theorem
-     has been used as a test on the quality of approximate
-     calculations. The complex virial theorem provides a similar
-     test for resonance calculations.

## 		Introduction

- _Debye Plasma_

-  The distance over which an electric field is
-    screened by redistribution of electrons and
-    ions in plasma.

- λd

- Fig 1:Debye Sphere

- -Test charge repels positive ions
- and attracts electrons, so around
- test charge electron density increases
- and ion  density  decreases and test
- charge gathers a shielding cloud.

- Q

- +e

- -e

- -e

- +e

- 		Introduction

##       Theoretical concepts:

- _Non-relativistic two electron Hamiltonian_

- Perimetric coordinate

-       Theoretical concepts:

- _Eigenvalue problem to solve_

## Results:

- Singlet P-and D- states energy

## Triplet P- and D- States Energy

-  For very large values of D the Debye screening potential converges toward the Coulomb potential

## Dilatation transformation

- Highly successful and most interesting approach so that bound state
- methods can be used for continuum problem.

- Solved for eigenvalues by diagonalisation

- The most attractive feature of the dilatation    transformation  approach is the relative ease with which standard programs of quantum chemistry can be modified to include also non-stationary systems such as autoionising states, electron scattering resonances.

## Resonances 

- Resonances
- hidden

- Continuum cut

- Resonances
- uncovered

- Bound states

- Spectrum of H(0)

- Spectrum of H(Ө)

- 2ReӨ

-  Bound state eigenvalue remained fixed at their untransformed values.
-  Continuous eigenvalues are rotated through an angle -2Ө.
-  When Ө is large enough, a new eigenvalue of H(Ө) appears
-   at the position of resonance.
-  Once uncovered ,this resonance eigenvalue is invariant to Ө until  it is covered up by another branch at which value of Ө it disappears.

## SUM-RULES

- Idea: Look at an already dilated complex energy eigenvalue and study how it
- reacts to an additional small dilatation !!!

**How to find good variational criteria for resonances in the dilatation method**

-              All coefficients of the various powers of iδ must vanish

- Second  order
- Sum rule

## Virial theorem

- -First member in the set of sum rules
- -Tool to obtain the correct balance of potential and kinetic energy
- -Virial theorem monitors the quality of  bound state calculations.
- Complex virial theorem supplies stationarity condition for resonances.
- The virial theorem  can be used for finite-basis-set calculations as a
-   way of picking out the values of rotation angle for which the eigenvalue
-   is nearly stable

## The Experimental situation

| E (ev) | Г (ev) | S1 | S2 |
| --- | --- | --- | --- |
| 57.84 | 0.127 | 0.7×10-6 | 0.178 |

- Results by Bhatia& Temkin

- E = 57.84
**Г**** = 0.124**

## Results:

- 0.42812×10-2

- -2.0260117

- 100

- Triplet D- state

- 0.47641×10-3

- -2.0259898

- 100

- Singlet D- state

- -0.2621986×10-1

- -1.9639068

- 20

- Triplet P- state

- -0.1141399×10-2

- -1.9813189

- 20

- Singlet P- state

- Sum rule satisfaction

-     Energy (calculated)
-                  a.u

- Debye Length
-       λd (a.u)

-             State

## Conclusions:

- Correlated wave functions for two-electron systems have widely been applied
- to calculate the bound P and D state energies of Helium atom in a screening
- environments and it is found that Debye screening potential converges toward
- the Coulomb potential in the limit of weak screening.
- .
- In all calculations the satisfaction of quantum virial theorem and related sum rules
- has been found to provide a high degree of stability.
- The degree of satisfaction of sum rules has been used as a measure for the quality
- of the particular resonance energy calculations.

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