Notebooks · lib/hfref/h_sto3g

One hydrogen atom, one electron

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).

Conventions:

atoms[natm,4] = (Z, x, y, z) in Bohr; shells[nsh,8] = (atom, l, nprim, p0, cart0, sph0, ncart, nsph); prims[nprim,2] = (exponent, raw bas_ctr_coeff). General contractions are split into segmented shells, and primitives whose coefficient is exactly zero are dropped.
_unit matrices are unit-Cartesian: PySCF's Cartesian integrals divided by F_i F_j, so every component has self-overlap 1. _sph matrices are PySCF's spherical ones, and _m() is M with X_sph = M^T X_unit M.
Packed-8 ERIs in chemist's notation: ij = i(i+1)/2 + j for i >= j, and ijkl = ij(ij+1)/2 + kl for ij >= kl. Full when the basis has at most 19 functions; otherwise the pair diagonal and 256 seeded samples. Both carry _SUMSQ = sum Gp^2 and _WSUM = sum w_k Gp_k, where w_k = s_k / (2^31 - 1), s_k = 16807 s_(k-1) mod (2^31 - 1) and s_0 = the module's _SEED_WSUM (20260923).
Energies: _E_<CFG>_DEFAULT is the state PySCF reaches from the minao guess, _E_<CFG>_STABLE the lowest state reached by restarting from the stability-analysis orbitals until the energy stops dropping, and _E_<CFG> repeats the default. Densities (total for RHF, alpha and beta otherwise) are in the same basis convention as the matrices beside them: a unit-Cartesian density is F_i F_j times PySCF's. A _stable_ density appears only when the stable state differs from the default.
In a module with both bases, spherical names carry no suffix and Cartesian ones end in _cart / _CART.

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[1]lines 36–485

# Seed of the LCG weights in every _WSUM checksum.
H_STO3G_SEED_WSUM = 20260923

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

#| Reshape a flat row-major vector into r x c.
def h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_pair_table(n) -> Arr {
    pr = zeros(n, n)
    for i to n {

# Seed of the LCG weights in every _WSUM checksum.
H_STO3G_SEED_WSUM = 20260923

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

#| Reshape a flat row-major vector into r x c.
def h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_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 h_sto3g_pair_table(n) -> Arr {
    pr = zeros(n, n)
    for i to n {