# Theoretical aspects

The many-electron equation to be solved is the Dirac-Coulomb-(Breit) equation for N electrons

(1)

(2)

Here h_{i} is the one-electron Dirac Hamiltonian for electron i

(3)

The inner product
is taken between the vector of three Pauli spin matrices and the momentum operator **p** (p_{x},p_{y},p_{z}), acting on the coordinates of electron i. In our notation 1_{2} is used for a 2x2 unit matrix. The scalar potential arises in this approximation from the fixed nuclear framework.
The program package uses atomic units in which the speed of light, c, is taken to be 137.0359895. In these units m, the mass of the electron, e, the elementary charge are set to 1.

To define a many-electron Hamiltonian one needs a two-electron operator g_{ij}. A correct relativistic two-electron operator cannot be written down in closed form. From the theory of Quantum Electro Dynamics one can derive a series expansion of the complete interaction [1]. The first term is the Coulomb interaction

(4)

The next term is the Breit [2] correction. Its magnetic part, the Gaunt [3] operator, is presently implemented in the program package

(5)

Here a is the vector of four-component Pauli matrices

(6)

With the two-electron operator defined as the Coulomb plus the Gaunt operator

(7)

the equations 1 and 2 define our basic physical starting point.

### References in Theory

- I.P.Grant and H.M.Quiney, Adv.At.Mol. Phys.
**23**, 37 (1988).
- G.Breit, Phys.Rev.
**34**, 553 (1929).
- J.A.Gaunt, Proc.Roy.Soc.
**A122**, 513 (1929).

MOLFDIR Contents