Dear friend Josip Jakovac

I am using my knowledge of LCAO to answer your question. I will read attached article today and will modify my answer if needed.

The LCAO (Linear Combination of Atomic Orbitals) method is a popular approach to obtain the wave functions of pi and pi* orbitals in graphene. In this method, the wave functions are expressed as a linear combination of atomic orbitals of the carbon atoms in the graphene lattice.

To begin, we need to consider the atomic orbitals of carbon atoms that form the graphene lattice. Each carbon atom has three sp2 hybrid orbitals and one unhybridized p orbital. These four orbitals participate in bonding and give rise to sigma (σ) and pi (π) molecular orbitals.

The sigma orbitals result from the overlap of sp2 hybrid orbitals of neighboring carbon atoms, while the pi orbitals arise from the overlap of unhybridized p orbitals. In graphene, there are two types of pi orbitals: pi (π) and pi* (π*) orbitals. The pi orbitals are bonding orbitals, while the pi* orbitals are antibonding orbitals.

To obtain the wave functions of pi and pi* orbitals using LCAO, we need to start by writing the atomic orbitals of each carbon atom in the graphene lattice. Then, we can express the wave function of the pi orbital as a linear combination of p orbitals of the neighboring carbon atoms. Similarly, the wave function of the pi* orbital can be expressed as a linear combination of p orbitals of the neighboring carbon atoms, but with a different sign in front of each term compared to the pi orbital.

The LCAO wave function for the pi orbital of graphene can be expressed as:

π(x,y) = ∑ Cn φn(x,y)

where Cn is a constant, and φn(x,y) is the wave function of the nth p orbital of the neighboring carbon atoms. The summation is taken over all neighboring carbon atoms.

Similarly, the LCAO wave function for the pi* orbital of graphene can be expressed as:

π*(x,y) = ∑ Cn φn*(x,y)

where φn*(x,y) is the wave function of the nth p orbital of the neighboring carbon atoms, but with a different sign compared to the pi orbital.

To calculate the constants Cn, we can use the method of linear variation, which minimizes the energy of the system by adjusting the values of the constants. The final wave functions obtained by LCAO will depend on the values of the constants and the choice of atomic orbitals used to construct the wave functions.

In summary, the LCAO method is a powerful tool to obtain the wave functions of pi and pi* orbitals in graphene. By expressing the wave functions as a linear combination of atomic orbitals, we can obtain a better understanding of the electronic structure and properties of graphene

We should be careful with the names here: talking about localized orbitals is accurate for isolated molecules, but for periodic materials like graphene talking about bands is more accurate although the math behind it is similar but with some extra steps.

You sound like you are looking for an analytic expression for the band wavefunctions. Unfortunately, that is not provided, you can get the coefficients which the basis functions have in the calculation result which is what the equations in Kaushik Shandilya 's answer represent, but if you want to have something to build upon for "pen-and-paper" math, that is not provided here.

Kaushik, thank you very much for making the idea behind LCAO clear to me. I have another question for you. You mention C pz atoms orbital wave functions. In Quantum mechanics, I learned how to calculate those for H atom (and alike ones). But unlike those, C atom has 4 valence electrons... Which function should I use here?

Jurgen, also thank you very much for your comment. Yep, "carbon orbitals" was "lapsus calami", induced by the fact that I went through numerous articles about LCAO, and none of those were as clear as Kaushik explained. Now, as I mentioned in the original post, I need wave functions of those. Although pen and paper method would be preferable, numerical stuff would also do fine for me. When doing PDOS, I have noticed that Quantum Espresso does LCAO method for calculation of the pz orbitals (among others). Could I somehow use these results to get the final result?

Thanks to both of you in advance,

Josip

Regarding the orbitals: most QC packages use GTOs (Gaussian Type Orbitals) instead of STOs (Slater Type Orbitals) because these are easier to handle for the computer. That means that instead of the normal exponential decay term in the function you have a Gaussian function. Depending on what basis set you use, there will be more or less redundant functions which are projected out by the calculation, so just because there is only a certain number of orbitals in "reality" [I mean, strictly spoken orbitals don't exist either] doesn't mean you have to use that exact number as a basis set size. If you use a "standard" combination like B3LYP/def2-TZVP, the "TZVP" stands for "triple zeta valence" which means that in the beginning there are three basis functions per valence orbital, actually.

Regarding the coefficients: yes, of course, the coefficients with respect to the basis set functions can be obtained, but I don't know how to do this for Quantum Espresso in particular.