Notebooks · lib/hfref/c2s

Cartesian to spherical, one shell at a time

Generated by tools/hf_reference.py from PySCF 2.5.0. Do not edit; rerun the generator. Settings: conv_tol 10⁻¹², init_guess minao, symmetry off, and each molecule built with cart=False and with cart=True. Every stored density is then re-converged from itself at conv_tol_grad 10⁻¹⁰ or tighter, and its _SELF_STEP constant says how far one PySCF step moves it (below 10⁻¹⁰).

M_l maps the unit-Cartesian components of one shell (PySCF order: lx from l down, then ly) to PySCF's spherical ones (p as x, y, z; d as xy, yz, z^2, xz, x^2 - y^2; f and g as m = -l..l): M_l = diag(F) cart2sph(l, 'sp'), with F^2 = 4 pi (2i-1)!! (2j-1)!! (2k-1)!! / (2l+1)!! for l >= 2 and F = 1 below.

Also the two constants pyscf/gto/test/test_mole.py:520 and :529 assert: the s and p factors of the unnormalized cart2sph, 1/(2 sqrt(pi)) and sqrt(3/(4 pi)).

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[1]lines 21–489

# ---- helpers for the tests below ----

#| Reshape a flat row-major vector into r x c.
def c2s_mat(v: Arr, r, c) -> Arr {
    a = zeros(r, c)
    for i to r {
        for j to c {
            a[i, j] = v[i * c + j]
        }
    }
    return a
}

def c2s_eye(n) -> Arr {
    a = zeros(n, n)
    for i to n {
        a[i, i] = 1.0
    }
    return a
}

#| Largest |A_ij - A_ji| / (1 + |A_ij|): a kinetic entry near 100 carries
#| rounding of about 1e-14, so the gap is measured relative to the entry.
def c2s_asym(a: Arr, n) {
    w = 0.0
    for i to n {
        for j to n {
            w = max(w, abs(a[i, j] - a[j, i]) / (1.0 + abs(a[i, j])))
        }
    }
    return w
}

#| Largest |A_ii - 1|.
def c2s_diag_dev(a: Arr, n) {
    w = 0.0
    for i to n {
        w = max(w, abs(a[i, i] - 1.0))
    }
    return w
}

#| sum_ij A_ij B_ij: Tr(D S) is the electron count.
def c2s_trace2(a: Arr, b: Arr, n) {
    t = 0.0
    for i to n {
        for j to n {
            t = t + a[i, j] * b[i, j]
        }
    }
    return t
}

#| pr[i, j] = the packed pair index of (i, j), in either order.
def c2s_pair_table(n) -> Arr {
    pr = zeros(n, n)
    for i to n {
        for j to i + 1 {
            pr[i, j] = i * (i + 1) / 2 + j
            pr[j, i] = pr[i, j]

# ---- helpers for the tests below ----

#| Reshape a flat row-major vector into r x c.
def c2s_mat(v: Arr, r, c) -> Arr {
    a = zeros(r, c)
    for i to r {
        for j to c {
            a[i, j] = v[i * c + j]
        }
    }
    return a
}

def c2s_eye(n) -> Arr {
    a = zeros(n, n)
    for i to n {
        a[i, i] = 1.0
    }
    return a
}

#| Largest |A_ij - A_ji| / (1 + |A_ij|): a kinetic entry near 100 carries
#| rounding of about 1e-14, so the gap is measured relative to the entry.
def c2s_asym(a: Arr, n) {
    w = 0.0
    for i to n {
        for j to n {
            w = max(w, abs(a[i, j] - a[j, i]) / (1.0 + abs(a[i, j])))
        }
    }
    return w
}

#| Largest |A_ii - 1|.
def c2s_diag_dev(a: Arr, n) {
    w = 0.0
    for i to n {
        w = max(w, abs(a[i, i] - 1.0))
    }
    return w
}

#| sum_ij A_ij B_ij: Tr(D S) is the electron count.
def c2s_trace2(a: Arr, b: Arr, n) {
    t = 0.0
    for i to n {
        for j to n {
            t = t + a[i, j] * b[i, j]
        }
    }
    return t
}

#| pr[i, j] = the packed pair index of (i, j), in either order.
def c2s_pair_table(n) -> Arr {
    pr = zeros(n, n)
    for i to n {
        for j to i + 1 {
            pr[i, j] = i * (i + 1) / 2 + j
            pr[j, i] = pr[i, j]