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## 4.1 Execution time

The following is a rough estimate of the complexity of a plain scf calculation with `pw.x`, for NCPP. USPP and PAW give raise additional terms to be calculated, that may add from a few percent up to 30-40% to execution time. For phonon calculations, each of the 3Nat modes requires a time of the same order of magnitude of self-consistent calculation in the same system (possibly times a small multiple). For `cp.x`, each time step takes something in the order of Th + Torth + Tsub defined below.

The time required for the self-consistent solution at fixed ionic positions, Tscf , is:

Tscf = NiterTiter + Tinit

where Niter = number of self-consistency iterations (`niter`), Titer = time for a single iteration, Tinit = initialization time (usually much smaller than the first term).

The time required for a single self-consistency iteration Titer is:

Titer = NkTdiag + Trho + Tscf

where Nk = number of k-points, Tdiag = time per Hamiltonian iterative diagonalization, Trho = time for charge density calculation, Tscf = time for Hartree and XC potential calculation.

The time for a Hamiltonian iterative diagonalization Tdiag is:

Tdiag = NhTh + Torth + Tsub

where Nh = number of Hψ products needed by iterative diagonalization, Th = time per Hψ product, Torth = CPU time for orthonormalization, Tsub = CPU time for subspace diagonalization.

The time Th required for a Hψ product is

Th = a1MN + a2MN1N2N3log(N1N2N3) + a3MPN.

The first term comes from the kinetic term and is usually much smaller than the others. The second and third terms come respectively from local and nonlocal potential. a1, a2, a3 are prefactors (i.e. small numbers (1)), M = number of valence bands (`nbnd`), N = number of PW (basis set dimension: `npw`), N1, N2, N3 = dimensions of the FFT grid for wavefunctions (`nr1s`, `nr2s`, `nr3s`; N1N2N3∼8N ), P = number of pseudopotential projectors, summed on all atoms, on all values of the angular momentum l, and m = 1,..., 2l + 1.

The time Torth required by orthonormalization is

Torth = b1NMx2

and the time Tsub required by subspace diagonalization is

Tsub = b2Mx3

where b1 and b2 are prefactors, Mx = number of trial wavefunctions (this will vary between M and 2÷4M, depending on the algorithm).

The time Trho for the calculation of charge density from wavefunctions is

Trho = c1MNr1Nr2Nr3log(Nr1Nr2Nr3) + c2MNr1Nr2Nr3 + Tus

where c1, c2, c3 are prefactors, Nr1, Nr2, Nr3 = dimensions of the FFT grid for charge density (`nr1`, `nr2`, `nr3`; Nr1Nr2Nr3∼8Ng, where Ng = number of G-vectors for the charge density, `ngm`), and Tus = time required by PAW/USPPs contribution (if any). Note that for NCPPs the FFT grids for charge and wavefunctions are the same.

The time Tscf for calculation of potential from charge density is

Tscf = d2Nr1Nr2Nr3 + d3Nr1Nr2Nr3log(Nr1Nr2Nr3)

where d1, d2 are prefactors.

For hybrid DFTs, the dominant term is by far the calculation of the nonlocal (Vxψ) product, taking as much as

Texx = eNkNqM2N1N2N3log(N1N2N3)

where Nq is the number of points in the k + q grid, determined by options `nqx1,nqx2,nqx3`, e is a prefactor.

The above estimates are for serial execution. In parallel execution, each contribution may scale in a different manner with the number of processors (see below).

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