Notebooks · posts/two-integral-families

Two methods compute water's 6-31G integrals

Adam Zettel · September 24, 2026 · written with an agent

McMurchie-Davidson and Head-Gordon-Pople compute the same 4186 electron repulsion integrals for water in the 6-31G basis by different recurrences. They agree to rounding, each matches PySCF, and each one alone gives PySCF's Hartree-Fock energy. With a third implementation in the comparison, the cross-check names the implementation that disagrees and the integral where it does.

(μν ∣ λσ) = \iint \frac{χ _{μ} ( 1 ) χ _{ν} ( 1 ) χ _{λ} ( 2 ) χ _{σ} ( 2 )}{r _{12}} d 1 d 2

The electron repulsion integrals are most of the cost of a Hartree-Fock calculation and the easiest place for a bug that does not announce itself. lib/hf computes them more than one way and compares the answers. This post takes the two methods the library cross-checks by default and runs each one alone on water in 6-31G: 13 basis functions and 4186 integrals after the eightfold permutation symmetry.

McMurchie-Davidson writes each product of two Gaussians as a sum of Hermite Gaussians. The integral becomes two matrix products over those expansions and the Hermite Coulomb integrals $R_{t uv}$ at the distance between the two product centers.

Head-Gordon-Pople starts from Obara and Saika's vertical recurrence, which builds up the angular momentum on one center of each pair, primitive by primitive. A horizontal recurrence then moves it onto the other center once per contracted shell quartet.

They share no recurrence. Both take the Boys function from lib/hf/boys, which is checked on its own against mpmath values (tools/boys_reference.py).

About 60 percent of the integrals come out as the same double from both methods, bit for bit: 2483 of the 4186. The exact count depends on the last bit of exp, so a different math library would change it. Every Viv engine, native or in the browser, uses the same one, so the count is the same in all of them, and a test checks it. The figure sorts the integrals that differ by how far apart they are, measured as $∣ a - b ∣/ (1 + ∣ a ∣)$ , the measure the library's cross-check uses. None is as far as $1 0^{- 15}$ . For the integrals larger than 0.01 the gap is at most 5.75 times the double-precision epsilon relative to the integral itself. The screened McMurchie-Davidson code skips any shell quartet whose Schwarz bound is under $1 0^{- 14}$ , so a few integrals no larger than $3 \times 1 0^{- 20}$ are exactly zero in its set.

Each method is also checked against PySCF's stored integrals within $1 0^{- 12}$ , and each one alone, fed to the self-consistent field with the default implementation of every other step, gives PySCF's energy $- 75.98394849810573$ within $1 0^{- 9}$ .

Finding which implementation is wrong. The cross-check takes the median of the implementations' answers, entry by entry, and blames the implementation farthest from it, at a named integral. For an even count the program takes the upper of the two middle values as the median. With two implementations that is the larger value, so a disagreement is detected but the blame falls on whichever one gave the lower value. With three, the median is a correct value. The last part of the program adds a third implementation that returns the Head-Gordon-Pople integrals with integral 2000, which is $- 0.117$ , multiplied by $1 - 1 0^{- 9}$ . That makes it larger, so with only two implementations the median is the wrong value and the correct one takes the blame. With three, the check names the altered implementation and integral 2000.

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[1]lines 61–239

from hf.hf import hf_gap, hf_hcore, hf_overlap, hf_scf, hf_setup_of
from hf.molecule import molecule_packs
from hf.basis import bas_nao_cart
from hf.md_eri import MDERI_SCREEN_TOL
from hf.eri_store import es_len8
from hf.opts import scf_defaults
from hf.integrals import int_eri, INT_TOL_ERI
from hf.sweep import sw_check3_arr, SW_REP_INDEX, sw_rep_new, SW_REP_NSTRAT, SW_REP_STRAT
from hfref.water_631g import (
    water631g_atoms, water631g_eri8_cart, WATER631G_ERI8_CART_LEN, water631g_prims,
    water631g_shells,
)

fam_atoms = water631g_atoms()
fam_shells = water631g_shells()
fam_prims = water631g_prims()
fam_n = bas_nao_cart(fam_shells)
fam_len = es_len8(fam_n)

# ---- each method alone: a mask with one bit set selects one implementation
md_report = sw_rep_new()
G_mcmurchie_davidson = int_eri(fam_atoms, fam_shells, fam_prims, 2, md_report)
hgp_report = sw_rep_new()
G_head_gordon_pople = int_eri(fam_atoms, fam_shells, fam_prims, 8, hgp_report)

# ---- how far apart they are, integral by integral, in decades
identical_integrals = 0
screened_to_zero = 0
# the decades of the histogram, 10^-22 to 10^-14
NDECADES = 8
decade_count = zeros(NDECADES)
worst_relative_gap = 0.0
smallest_gap = 1.0
# the largest gap among integrals above 0.01, in units of the double epsilon
# relative to the integral, and the largest integral screening set to zero
worst_large_gap_eps = 0.0
largest_screened = 0.0
for i to fam_len {
    a = G_mcmurchie_davidson[i]
    b = G_head_gordon_pople[i]
    d = abs(a - b) / (1.0 + abs(a))
    worst_relative_gap = max(worst_relative_gap, d)
    if abs(a) > 0.01 {
        worst_large_gap_eps = max(worst_large_gap_eps, abs(a - b) / abs(a) / 2.220446049250313e-16)
    }
    if a == 0.0 {
        largest_screened = max(largest_screened, abs(b))
    }
    if d == 0.0 {
        identical_integrals = identical_integrals + 1
    }
    else {
        smallest_gap = min(smallest_gap, d)
        if a == 0.0 {
            screened_to_zero = screened_to_zero + 1
        }
        # decade k holds 10^(-22+k) <= d < 10^(-21+k)
        k = floor(log(d) / log(10.0)) + 22
        k = min(max(k, 0), NDECADES - 1)
        decade_count[k] = decade_count[k] + 1
    }
}
say("integrals after the eightfold symmetry", fam_len)

decade_center = zeros(NDECADES)
for k to NDECADES {
    decade_center[k] = -21.5 + k
}
figure("Integrals where McMurchie-Davidson and Head-Gordon-Pople differ, by size of the difference", "$\\log_{10}$ of $|a - b|/(1 + |a|)$", "number of integrals")
bars(decade_center, decade_count, "the integrals that are not bit-identical")

# ---- each method alone through the SCF, with the default implementation of every other step
fam_mol = molecule_packs(fam_atoms, fam_shells, fam_prims)
S_water = hf_overlap(fam_mol, scf_defaults())
H_water = hf_hcore(fam_mol, scf_defaults())
E_mcmurchie_davidson = hf_scf(hf_setup_of(fam_mol, H_water, S_water, G_mcmurchie_davidson, scf_defaults())).energy
E_head_gordon_pople = hf_scf(hf_setup_of(fam_mol, H_water, S_water, G_head_gordon_pople, scf_defaults())).energy
show E_mcmurchie_davidson
show E_head_gordon_pople

# ---- a planted error, and the cross-check that names it
# Each implementation receives the two integral sets already computed above,
# plus the change to make [index, factor]; the cross-check sees only the arrays
# they return.
NUDGE = [2000, 0.999999999]

#| Implementation 0 for the cross-check: the McMurchie-Davidson integrals, unchanged.
def fam_first(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    return g_md + 0.0
}

#| Implementation 1: the Head-Gordon-Pople integrals, unchanged.
def fam_second(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    return g_hgp + 0.0
}

#| The deliberately wrong implementation: the Head-Gordon-Pople integrals with
#| integral `nudge[0]` multiplied by `nudge[1]`.
def fam_second_nudged(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    g = g_hgp + 0.0
    g[nudge[0]] = g[nudge[0]] * nudge[1]
    return g
}

#| Run the cross-check over the implementations in `ms`; returns
#| `[verdict, index of the implementation blamed, index of the integral]`.
def fam_blame(ms, g_md: Arr, g_hgp: Arr, nudge: Arr, n) {
    rep = sw_rep_new()
    out = zeros(n)
    v = sw_check3_arr(ms, g_md, g_hgp, nudge, n, n, INT_TOL_ERI, rep, out)
    return [v, rep[SW_REP_STRAT], rep[SW_REP_INDEX]]
}

three_members = fam_blame([fam_first, fam_second, fam_second_nudged], G_mcmurchie_davidson, G_head_gordon_pople, NUDGE, fam_len)
two_members = fam_blame([fam_first, fam_second_nudged], G_mcmurchie_davidson, G_head_gordon_pople, NUDGE, fam_len)
blamed_member_of_three = three_members[1]
blamed_integral_of_three = three_members[2]
say("with three implementations, the one blamed (2 is the altered one)", blamed_member_of_three)
say("the integral it is blamed for", blamed_integral_of_three)

test "each method alone matches PySCF's 4186 integrals within 1e-12, and the two agree to 1e-15" {
    assert_near(fam_len, 4186, 0.0)
    assert_near(md_report[SW_REP_NSTRAT], 1, 0.0)
    assert_near(hgp_report[SW_REP_NSTRAT], 1, 0.0)
    assert_lt(hf_gap(G_mcmurchie_davidson, water631g_eri8_cart(), WATER631G_ERI8_CART_LEN), 1.0e-12)
    assert_lt(hf_gap(G_head_gordon_pople, water631g_eri8_cart(), WATER631G_ERI8_CART_LEN), 1.0e-12)
    assert_lt(worst_relative_gap, 1.0e-15)
    # the decades span every gap, so no count was clamped into an end bin
    assert_lt(1.0e-22, smallest_gap)
    # screening zeroes a few tiny integrals; never more than a handful
    assert_lt(screened_to_zero, 10)
    # every integral is either identical or counted in one decade
    total = identical_integrals
    for k to NDECADES {
        total = total + decade_count[k]
    }
    assert_near(total, fam_len, 0.0)
}

test "2483 integrals are bit-identical; the rest are within 5.75 epsilon or screened below 3e-20" {
    assert_near(identical_integrals, 2483, 0.0)
    assert_lt(5.7, worst_large_gap_eps)
    assert_lt(worst_large_gap_eps, 5.75)
    assert_lt(largest_screened, 3.0e-20)
}

test "each method alone gives PySCF's RHF energy -75.98394849810573 within 1e-9" {
    assert_near(E_mcmurchie_davidson, -75.98394849810573, 1.0e-9)
    assert_near(E_head_gordon_pople, -75.98394849810573, 1.0e-9)
    assert_lt(abs(E_mcmurchie_davidson - E_head_gordon_pople), 1.0e-11)
}

test "with three implementations the check names the altered one and integral 2000; with two it detects the error but blames the correct one" {
    assert_near(three_members[0], 0, 0.0)
    assert_near(blamed_member_of_three, 2, 0.0)
    assert_near(blamed_integral_of_three, 2000, 0.0)
    # with two, the median is the larger value, the altered one, so the blame lands on implementation 0
    assert_near(two_members[0], 0, 0.0)
    assert_near(two_members[1], 0, 0.0)
    assert_near(two_members[2], 2000, 0.0)
}
test "water in 6-31G has 13 basis functions and 4186 integrals after the eightfold symmetry" {
    assert_near(fam_n, 13, 0.0)
    assert_near(fam_len, 4186, 0.0)
}
test "about 60 percent of the integrals are bit-identical from the two methods" {
    # 2483 of 4186 is 59.3 percent; "about 60" allows half of the stated 10
    assert_near(identical_integrals / fam_len, 0.60, 0.05)
}
test "the screened McMurchie-Davidson code skips quartets whose Schwarz bound is under 1e-14" {
    assert_near(MDERI_SCREEN_TOL, 1.0e-14, 0.0)
}
test "the integral the wrong implementation alters, integral 2000, is -0.117" {
    assert_near(G_head_gordon_pople[NUDGE[0]], -0.117, 0.0005)
    assert_near(NUDGE[1], 1.0 - 1.0e-9, 1.0e-16)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

from hf.hf import hf_gap, hf_hcore, hf_overlap, hf_scf, hf_setup_of
from hf.molecule import molecule_packs
from hf.basis import bas_nao_cart
from hf.md_eri import MDERI_SCREEN_TOL
from hf.eri_store import es_len8
from hf.opts import scf_defaults
from hf.integrals import int_eri, INT_TOL_ERI
from hf.sweep import sw_check3_arr, SW_REP_INDEX, sw_rep_new, SW_REP_NSTRAT, SW_REP_STRAT
from hfref.water_631g import (
    water631g_atoms, water631g_eri8_cart, WATER631G_ERI8_CART_LEN, water631g_prims,
    water631g_shells,
)

fam_atoms = water631g_atoms()
fam_shells = water631g_shells()
fam_prims = water631g_prims()
fam_n = bas_nao_cart(fam_shells)
fam_len = es_len8(fam_n)

# ---- each method alone: a mask with one bit set selects one implementation
md_report = sw_rep_new()
G_mcmurchie_davidson = int_eri(fam_atoms, fam_shells, fam_prims, 2, md_report)
hgp_report = sw_rep_new()
G_head_gordon_pople = int_eri(fam_atoms, fam_shells, fam_prims, 8, hgp_report)

# ---- how far apart they are, integral by integral, in decades
identical_integrals = 0
screened_to_zero = 0
# the decades of the histogram, 10^-22 to 10^-14
NDECADES = 8
decade_count = zeros(NDECADES)
worst_relative_gap = 0.0
smallest_gap = 1.0
# the largest gap among integrals above 0.01, in units of the double epsilon
# relative to the integral, and the largest integral screening set to zero
worst_large_gap_eps = 0.0
largest_screened = 0.0
for i to fam_len {
    a = G_mcmurchie_davidson[i]
    b = G_head_gordon_pople[i]
    d = abs(a - b) / (1.0 + abs(a))
    worst_relative_gap = max(worst_relative_gap, d)
    if abs(a) > 0.01 {
        worst_large_gap_eps = max(worst_large_gap_eps, abs(a - b) / abs(a) / 2.220446049250313e-16)
    }
    if a == 0.0 {
        largest_screened = max(largest_screened, abs(b))
    }
    if d == 0.0 {
        identical_integrals = identical_integrals + 1
    }
    else {
        smallest_gap = min(smallest_gap, d)
        if a == 0.0 {
            screened_to_zero = screened_to_zero + 1
        }
        # decade k holds 10^(-22+k) <= d < 10^(-21+k)
        k = floor(log(d) / log(10.0)) + 22
        k = min(max(k, 0), NDECADES - 1)
        decade_count[k] = decade_count[k] + 1
    }
}
say("integrals after the eightfold symmetry", fam_len)

decade_center = zeros(NDECADES)
for k to NDECADES {
    decade_center[k] = -21.5 + k
}
figure("Integrals where McMurchie-Davidson and Head-Gordon-Pople differ, by size of the difference", "$\\log_{10}$ of $|a - b|/(1 + |a|)$", "number of integrals")
bars(decade_center, decade_count, "the integrals that are not bit-identical")

# ---- each method alone through the SCF, with the default implementation of every other step
fam_mol = molecule_packs(fam_atoms, fam_shells, fam_prims)
S_water = hf_overlap(fam_mol, scf_defaults())
H_water = hf_hcore(fam_mol, scf_defaults())
E_mcmurchie_davidson = hf_scf(hf_setup_of(fam_mol, H_water, S_water, G_mcmurchie_davidson, scf_defaults())).energy
E_head_gordon_pople = hf_scf(hf_setup_of(fam_mol, H_water, S_water, G_head_gordon_pople, scf_defaults())).energy
show E_mcmurchie_davidson
show E_head_gordon_pople

# ---- a planted error, and the cross-check that names it
# Each implementation receives the two integral sets already computed above,
# plus the change to make [index, factor]; the cross-check sees only the arrays
# they return.
NUDGE = [2000, 0.999999999]

#| Implementation 0 for the cross-check: the McMurchie-Davidson integrals, unchanged.
def fam_first(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    return g_md + 0.0
}

#| Implementation 1: the Head-Gordon-Pople integrals, unchanged.
def fam_second(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    return g_hgp + 0.0
}

#| The deliberately wrong implementation: the Head-Gordon-Pople integrals with
#| integral `nudge[0]` multiplied by `nudge[1]`.
def fam_second_nudged(g_md: Arr, g_hgp: Arr, nudge: Arr) -> Arr {
    g = g_hgp + 0.0
    g[nudge[0]] = g[nudge[0]] * nudge[1]
    return g
}

#| Run the cross-check over the implementations in `ms`; returns
#| `[verdict, index of the implementation blamed, index of the integral]`.
def fam_blame(ms, g_md: Arr, g_hgp: Arr, nudge: Arr, n) {
    rep = sw_rep_new()
    out = zeros(n)
    v = sw_check3_arr(ms, g_md, g_hgp, nudge, n, n, INT_TOL_ERI, rep, out)
    return [v, rep[SW_REP_STRAT], rep[SW_REP_INDEX]]
}

three_members = fam_blame([fam_first, fam_second, fam_second_nudged], G_mcmurchie_davidson, G_head_gordon_pople, NUDGE, fam_len)
two_members = fam_blame([fam_first, fam_second_nudged], G_mcmurchie_davidson, G_head_gordon_pople, NUDGE, fam_len)
blamed_member_of_three = three_members[1]
blamed_integral_of_three = three_members[2]
say("with three implementations, the one blamed (2 is the altered one)", blamed_member_of_three)
say("the integral it is blamed for", blamed_integral_of_three)

test "each method alone matches PySCF's 4186 integrals within 1e-12, and the two agree to 1e-15" {
    assert_near(fam_len, 4186, 0.0)
    assert_near(md_report[SW_REP_NSTRAT], 1, 0.0)
    assert_near(hgp_report[SW_REP_NSTRAT], 1, 0.0)
    assert_lt(hf_gap(G_mcmurchie_davidson, water631g_eri8_cart(), WATER631G_ERI8_CART_LEN), 1.0e-12)
    assert_lt(hf_gap(G_head_gordon_pople, water631g_eri8_cart(), WATER631G_ERI8_CART_LEN), 1.0e-12)
    assert_lt(worst_relative_gap, 1.0e-15)
    # the decades span every gap, so no count was clamped into an end bin
    assert_lt(1.0e-22, smallest_gap)
    # screening zeroes a few tiny integrals; never more than a handful
    assert_lt(screened_to_zero, 10)
    # every integral is either identical or counted in one decade
    total = identical_integrals
    for k to NDECADES {
        total = total + decade_count[k]
    }
    assert_near(total, fam_len, 0.0)
}

test "2483 integrals are bit-identical; the rest are within 5.75 epsilon or screened below 3e-20" {
    assert_near(identical_integrals, 2483, 0.0)
    assert_lt(5.7, worst_large_gap_eps)
    assert_lt(worst_large_gap_eps, 5.75)
    assert_lt(largest_screened, 3.0e-20)
}

test "each method alone gives PySCF's RHF energy -75.98394849810573 within 1e-9" {
    assert_near(E_mcmurchie_davidson, -75.98394849810573, 1.0e-9)
    assert_near(E_head_gordon_pople, -75.98394849810573, 1.0e-9)
    assert_lt(abs(E_mcmurchie_davidson - E_head_gordon_pople), 1.0e-11)
}

test "with three implementations the check names the altered one and integral 2000; with two it detects the error but blames the correct one" {
    assert_near(three_members[0], 0, 0.0)
    assert_near(blamed_member_of_three, 2, 0.0)
    assert_near(blamed_integral_of_three, 2000, 0.0)
    # with two, the median is the larger value, the altered one, so the blame lands on implementation 0
    assert_near(two_members[0], 0, 0.0)
    assert_near(two_members[1], 0, 0.0)
    assert_near(two_members[2], 2000, 0.0)
}
test "water in 6-31G has 13 basis functions and 4186 integrals after the eightfold symmetry" {
    assert_near(fam_n, 13, 0.0)
    assert_near(fam_len, 4186, 0.0)
}
test "about 60 percent of the integrals are bit-identical from the two methods" {
    # 2483 of 4186 is 59.3 percent; "about 60" allows half of the stated 10
    assert_near(identical_integrals / fam_len, 0.60, 0.05)
}
test "the screened McMurchie-Davidson code skips quartets whose Schwarz bound is under 1e-14" {
    assert_near(MDERI_SCREEN_TOL, 1.0e-14, 0.0)
}
test "the integral the wrong implementation alters, integral 2000, is -0.117" {
    assert_near(G_head_gordon_pople[NUDGE[0]], -0.117, 0.0005)
    assert_near(NUDGE[1], 1.0 - 1.0e-9, 1.0e-16)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors