background Theoretical Background

In Siesta, the wave-functions are expanded in localized orbitals:

\[|\Psi_i> = \sum_{\mu} { c^i_{\mu} |\mu>}\]

The coefficients \(c^i_{\mu}\) are produced by the diagonalization routines in Siesta and stored in the *.WFSX files.

Note

The above equation needs to be generalized to extended systems (k-points, phases, etc). This is shown in the the bands tutorial

The electrons will fill up the states up to the Fermi level. If we denote the occupation of a state i by \(f_i\), we can write for the total band-structure energy:

\[E_{BS} = \sum_i { f_i <\Psi_i|H|\Psi_i>} = \sum_i { f_i \sum_{\mu\nu} { c^i_{\mu}c^i_{\nu} H_{\mu\nu} } }\]

where \(H_{\mu\nu} = <\mu|H|\nu>\) are the matrix elements of the Hamiltonian in the basis set. Alternatively:

\[E_{BS} = \sum_i { f_i <\Psi_i|H|\Psi_i>} = \sum_i { f_i \sum_{\mu\nu} { \varepsilon_i c^i_{\mu}c^i_{\nu} S_{\mu\nu}}}\]

where \(S_{\mu\nu} = <\mu|\nu>\) are the elements of the overlap matrix, which is needed in the case of non-orthogonal orbitals.

In the same way, the total number of electrons is given formally by:

(1)\[N = \sum_i { f_i <\Psi_i|\Psi_i>} = \sum_i { f_i \sum_{\mu\nu} { c^i_{\mu}c^i_{\nu} S_{\mu\nu}}}\]

Mulliken charges

The last equation can be re-arranged as:

\[N = \sum_I { \sum_{\mu\in I} { \sum_{\nu} { \sum_i { f_i c^i_{\mu}c^i_{\nu} S_{\mu\nu}}}}}\]

where we have partitioned the \(\mu\) index in different subsets for different atoms I. It is now a small logical step to think of

\[q_I = \sum_{\mu\in I} { \sum_{\nu} { \sum_i { f_i c^i_{\mu}c^i_{\nu} S_{\mu\nu}}}}\]

as the “contribution of atom I” to the total charge of the system. These are the “Mulliken populations”, and are covered in this tutorial section

DOS and projected DOS

Now we can take a seemingly crazy step and write the occupation numbers in the above equations as:

(2)\[f_i = \int_0^{\varepsilon_F} { \delta(\varepsilon-\varepsilon_i) d\varepsilon }\]

since the integral will be one for states below the fermi level \(\varepsilon_F\) and zero for those above it. Then, the decomposition of N reads:

\[N = \int_0^{\varepsilon_F} { d\varepsilon \sum_i { \sum_{\mu} { \sum_{\nu} { c_{i\mu}c_{i\nu} S_{\mu\nu} \delta(\varepsilon - \varepsilon_i) } } } }\]

This way to write N is actually quite useful. Begin by taking out the integral sign:

\[g(\varepsilon) = \sum_i { \sum_{\mu} { \sum_{\nu} { c_{i\mu}c_{i\nu} S_{\mu\nu} \delta(\varepsilon - \varepsilon_i) } } }\]

This is the density of states, which can also be written more simply as

\[g(\varepsilon) = \sum_i { \delta(\varepsilon - \varepsilon_i) }\]

since the sum over \(\mu\nu\) is one for each i (exercise to the reader). More on the DOS in Siesta in this tutorial section

In the next stage, we take out the sum over \(\mu\), and get:

\[g_{\mu} (\varepsilon) = \sum_i { \sum_{\nu} { c_{i\mu}c_{i\nu} S_{\mu\nu} \delta(\varepsilon - \varepsilon_i) } }\]

which is the projected density of states (pDOS) over orbital \(\mu\). Note that when the basis is not orthogonal, the sum over \(\mu\) and the overlap factor are needed.

The tools available in Siesta to compute and process the DOS and pDOS are discussed in this how-to, and further tutorial exercises are in this tutorial section

COOP and COHP curves

Yes!. We can take out another sum in the above equation:

\[g_{\mu\nu} (\varepsilon) = \sum_i { c_{i\mu}c_{i\nu} S_{\mu\nu} \delta(\varepsilon - \varepsilon_i) }\]

This, if we follow the chain of reasoning, must be the contribution of the pair of orbitals \(\mu, \nu\), to the electron charge “at some energy”. If we had started from the formal decomposition of the band-structure energy, we would have arrived at:

\[h_{\mu\nu} (\varepsilon) = \sum_i { c_{i\mu}c_{i\nu} H_{\mu\nu} \delta(\varepsilon - \varepsilon_i) }\]

which similarly would be the pair contribution to the (band structure) energy.

The above two equations define what are known, respectively, as the Crystal Orbital Overlap Population (COOP) and Crystal Orbital Hamilton Population (COHP) curves. (Curves because they are functions of energy).

It turns out that one can obtain a lot of insight into the chemical bonding from these curves.

For background and examples, see the LOBSTER site for more information. LOBSTER is a program created by the group of Richard Dronskowski, one of the pioneers of this kind of bonding analysis in solid-state chemistry.

In the tutorial on COOP/COHP we offer a couple of examples to show the implementation in Siesta of this analysis machinery.