Notebooks · lib/hf/cart2sph

From Cartesian to spherical Gaussians, one shell at a time

The integral kernels in this library work on Cartesian components: a d shell has six (xx, xy, xz, yy, yz, zz) and an f shell ten. Most basis sets are defined on spherical components instead: five d functions and seven f. The sixth Cartesian d combination, xx + yy + zz = r^2, is an s function in disguise, and a spherical basis leaves it out. That is why water in 6-31G* has 19 Cartesian functions and 18 spherical ones.

Each spherical function of a shell is a fixed linear combination of the shell's Cartesian components, with the same radial contraction. Collect the coefficients in a matrix M_l, one row per unit-normalized Cartesian component (PySCF's order) and one column per spherical function:

spherical_m = sum_c M_l[c, m] * unit_c

Then any one-electron matrix goes over as X_sph = M^T X M, and the electron repulsion integrals get one factor of M on each of their four indices. Here M is the block-diagonal matrix for a whole basis, with one M_l block per shell. This module builds M_l and M; lib/hf/transform applies them.

Two constructions. c2s_closed(l) writes the table out by hand for l <= 3, from the textbook real polynomials:

d:  xy, yz, (2z^2 - x^2 - y^2)/2, xz, (x^2 - y^2)
f:  y(3x^2 - y^2), xyz, y(4z^2 - x^2 - y^2), z(2z^2 - 3x^2 - 3y^2),
    x(4z^2 - x^2 - y^2), z(x^2 - y^2), x(x^2 - 3y^2)

A polynomial sum_c a_c x^lx y^ly z^lz becomes a combination of unit components with coefficient a_c sqrt((2lx-1)!! (2ly-1)!! (2lz-1)!!), since a unit component carries the factor Nc = 1/sqrt(...) (see lib/hf/basis). Scaling each column to unit norm gives the numbers in the table. For the f function z(2z^2 - 3x^2 - 3y^2), for example, the squared norm is 60 in units of a unit component's self-overlap, so the zzz entry is 2 sqrt(15) / sqrt(60) = 1 and the xxz entry is -3 sqrt(3) / sqrt(60).

c2s_general(l) works for any l. It sums the general expansion of the real solid harmonics (Helgaker, Jorgensen and Olsen, Molecular Electronic-Structure Theory, eq. 6.4.47) into monomial coefficients, and then scales each column so that M_l^T G M_l = I, where G is the overlap of the unit components of one shell (c2s_unit_gram). The two constructions share no numbers, so their agreement at l <= 3 checks both, and the general one checks lib/hfref/c2s at l = 4.

Order and sign follow PySCF: p stays x, y, z (not m = -1, 0, 1); d is xy, yz, z^2, xz, x^2 - y^2; f and higher run m = -l .. l. The signs of the solid-harmonic expansion match PySCF's for every l checked (0 to 4).

Used with import hf.cart2sph, after import hf.basis.

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[1]lines 55–344

from hf.basis import (
    bas_nao_cart, bas_nao_sph, bas_nc, bas_ncart, BAS_NDIM, bas_nsph, bas_odfact, bas_powers,
    shell_cart0, shell_l, shell_sph0,
)

# c2s_closed writes M_l out by hand up to f; above it defers to c2s_general
C2S_CLOSED_LMAX = 3

#| n choose k, exact in doubles for the small arguments used here.
def c2s_binom(n, k) {
    if k < 0 or k > n {
        return 0.0
    }
    r = 1.0
    for i to k {
        r = r * (n - i) / (i + 1)
    }
    return round(r)
}

#| The overlap of the unit Cartesian components of one shell, `[ncart, ncart]`,
#| when they share one radial part. Per axis the Gaussian moment gives
#| `(p + q - 1)!!` for powers p and q with p + q even, and 0 when it is odd;
#| the unit factors `Nc` scale the result so the diagonal is 1.
def c2s_unit_gram(l) -> Arr {
    P = bas_powers(l)
    nc = bas_ncart(l)
    G = zeros(nc, nc)
    #: unit-component overlap: zero for an odd power on any axis, else (2k - 1)!!
    for i to nc {
        for j to nc {
            v = 1.0
            for d to BAS_NDIM {
                s = P[i, d] + P[j, d]
                if mod(s, 2.0) > 0.5 {
                    v = 0.0
                }
                else {
                    v = v * bas_odfact(s / 2)
                }
            }
            G[i, j] = v * bas_nc(P[i, 0], P[i, 1], P[i, 2]) * bas_nc(P[j, 0], P[j, 1], P[j, 2])
        }
    }
    return G
}

#| The row of component (lx, ly, lz) in PySCF's Cartesian order: the
#| components with a larger lx come first, and within one lx a larger ly.
def c2s_cart_row(l, lx, ly) {
    a = l - lx
    return a * (a + 1) / 2 + (a - ly)
}

#| M_l by hand for l <= 3. Rows are the unit Cartesian components in PySCF's
#| order, columns the spherical functions in PySCF's order. Each entry is a
#| coefficient of the polynomial named above it times
#| sqrt((2lx-1)!! (2ly-1)!! (2lz-1)!!), divided by the polynomial's norm.
#| Above l = 3 it returns `c2s_general(l)`.
def c2s_closed(l) -> Arr {
    if l > C2S_CLOSED_LMAX {
        return c2s_general(l)
    }
    M = zeros(bas_ncart(l), bas_nsph(l))
    if l == 0 {
        M[0, 0] = 1.0
    }
    if l == 1 {
        # x, y, z map to themselves
        M[0, 0] = 1.0
        M[1, 1] = 1.0
        M[2, 2] = 1.0
    }
    if l == 2 {
        # rows xx xy xz yy yz zz; columns xy yz z2 xz x2-y2
        # xy, yz, xz: one component each
        M[1, 0] = 1.0
        M[4, 1] = 1.0
        M[2, 3] = 1.0
        # (2z^2 - x^2 - y^2)/2 is sqrt(3) (zz - xx/2 - yy/2) in unit components,
        # and zz - xx/2 - yy/2 has squared norm 3/2 - 2/3 + 1/6 = 1 because
        # two different unit squares overlap by 1/3
        M[5, 2] = 1.0
        M[0, 2] = -0.5
        M[3, 2] = -0.5
        # x^2 - y^2: unit coefficients +-sqrt(3), squared norm 3 + 3 - 2 = 4
        M[0, 4] = sqrt(3.0) / 2.0
        M[3, 4] = -sqrt(3.0) / 2.0
    }
    if l == 3 {
        # rows xxx xxy xxz xyy xyz xzz yyy yyz yzz zzz; columns m = -3 .. 3
        # m = -3, y(3x^2 - y^2)
        M[1, 0] = 3.0 / (2.0 * sqrt(2.0))
        M[6, 0] = -sqrt(10.0) / 4.0
        # m = -2, xyz
        M[4, 1] = 1.0
        # m = -1, y(4z^2 - x^2 - y^2)
        M[8, 2] = sqrt(6.0 / 5.0)
        M[1, 2] = -sqrt(3.0 / 40.0)
        M[6, 2] = -sqrt(6.0) / 4.0
        # m = 0, z(2z^2 - 3x^2 - 3y^2)
        M[9, 3] = 1.0
        M[2, 3] = -3.0 / (2.0 * sqrt(5.0))
        M[7, 3] = -3.0 / (2.0 * sqrt(5.0))
        # m = 1, x(4z^2 - x^2 - y^2)
        M[5, 4] = sqrt(6.0 / 5.0)
        M[0, 4] = -sqrt(6.0) / 4.0
        M[3, 4] = -sqrt(3.0 / 40.0)
        # m = 2, z(x^2 - y^2)
        M[2, 5] = sqrt(3.0) / 2.0
        M[7, 5] = -sqrt(3.0) / 2.0
        # m = 3, x(x^2 - 3y^2)
        M[0, 6] = sqrt(10.0) / 4.0
        M[3, 6] = -3.0 / (2.0 * sqrt(2.0))
    }
    return M
}

#| The column of the spherical function with index m in PySCF's order. For p
#| PySCF keeps x, y, z, which are m = 1, -1, 0.
def c2s_sph_col(l, m) {
    if l == 1 {
        if m == 1 {
            return 0
        }
        if m == -1 {
            return 1
        }
        return 2
    }
    return m + l
}

#| M_l for any l from the real solid harmonic expansion
#|
#|     S_lm ~ sum_t sum_u sum_w (-1)^(t + (w - wm)/2) 4^-t C(l,t) C(l-t,|m|+t) C(t,u) C(|m|,w)
#|            x^(2t + |m| - 2u - w) y^(2u + w) z^(l - 2t - |m|)
#|
#| with t = 0 .. (l - |m|)/2, u = 0 .. t, and w running over the integers up
#| to |m| that are even for m >= 0 and odd for m < 0 (wm is 0 or 1 to
#| match). This is Helgaker's sum with v = w/2. The overall constant is left
#| out: each column is scaled afterwards to unit norm in the metric of
#| `c2s_unit_gram`, which also converts monomial coefficients to unit ones.
def c2s_general(l) -> Arr {
    nc = bas_ncart(l)
    ns = bas_nsph(l)
    A = zeros(nc, ns)
    #: solid harmonic's monomial coefficients, moved to unit components
    for mi to ns {
        m = mi - l
        am = abs(m)
        wm = 0
        if m < 0 {
            wm = 1
        }
        col = c2s_sph_col(l, m)
        for t to floor((l - am) / 2) + 1 {
            for u to t + 1 {
                w = wm
                while w <= am {
                    sgn = 1.0
                    if mod(t + (w - wm) / 2, 2.0) > 0.5 {
                        sgn = -1.0
                    }
                    c = sgn * pow(0.25, t) * c2s_binom(l, t) * c2s_binom(l - t, am + t) * c2s_binom(t, u) * c2s_binom(am, w)
                    lx = 2 * t + am - 2 * u - w
                    ly = 2 * u + w
                    row = c2s_cart_row(l, lx, ly)
                    # a monomial coefficient becomes a unit-component one by
                    # dividing by that component's Nc
                    A[row, col] = A[row, col] + c / bas_nc(lx, ly, l - lx - ly)
                    w = w + 2
                }
            }
        }
    }
    G = c2s_unit_gram(l)
    for j to ns {
        # the column's squared norm in the unit metric, a^T G a, entry by entry
        v = A[:, j]
        W = zeros(nc, nc)
        for a to nc {
            W[a, :] = v[a] * G[a, :] * v
        }
        A[:, j] = v * (1.0 / sqrt(sum(W)))
    }
    return A
}

#| The production M_l: the hand-written table where it exists, the general
#| construction above it.
def c2s_block(l) -> Arr {
    if l <= C2S_CLOSED_LMAX {

        return c2s_closed(l)
    }
    return c2s_general(l)
}

#| The block-diagonal M for a basis, `[nao_cart, nao_sph]`: shell s puts its
#| M_l at rows SH_CART0 and columns SH_SPH0, and everything between shells is
#| zero. Then PySCF's spherical X is `M^T X_unit M`.
def c2s_basis(shells: Arr) -> Arr {
    M = zeros(bas_nao_cart(shells), bas_nao_sph(shells))
    for s to len(shells) {
        l = shell_l(shells, s)
        B = c2s_block(l)
        c0 = shell_cart0(shells, s)
        s0 = shell_sph0(shells, s)
        M[c0:c0 + bas_ncart(l), s0:s0 + bas_nsph(l)] = B
    }
    return M
}

# ---- test helpers ----

#| The largest entry of abs(M^T G M - I) for one block.
def c2s_orth_gap(M: Arr, l) {
    E = transpose(M) @ c2s_unit_gram(l) @ M
    for i to len(E) {
        E[i, i] = E[i, i] - 1.0
    }
    return max(abs(E))
}

test "p is x, y, z and d is xy, yz, z2, xz, x2-y2" {
    # p: the identity, so column 0 is x, 1 is y, 2 is z
    c2s_tp = c2s_general(1)
    for c2s_ti to shape(c2s_tp)[0] {
        for c2s_tj to shape(c2s_tp)[1] {
            c2s_tw = 0.0
            if c2s_ti == c2s_tj {
                c2s_tw = 1.0
            }
            assert_near(c2s_tp[c2s_ti, c2s_tj], c2s_tw, 1.0e-15)
        }
    }
    # d rows: xx 0, xy 1, xz 2, yy 3, yz 4, zz 5
    c2s_td = c2s_general(2)
    # column 0 is xy alone, 1 is yz alone, 3 is xz alone
    assert_near(c2s_td[1, 0], 1.0, 1.0e-15)
    assert_near(c2s_td[4, 1], 1.0, 1.0e-15)
    assert_near(c2s_td[2, 3], 1.0, 1.0e-15)
    # column 2 is z^2 - (x^2 + y^2)/2, column 4 is x^2 - y^2
    assert_near(c2s_td[5, 2], 1.0, 1.0e-15)
    assert_near(c2s_td[0, 2], -0.5, 1.0e-15)
    assert_near(c2s_td[3, 2], -0.5, 1.0e-15)
    assert_near(c2s_td[0, 4], sqrt(3.0) / 2.0, 1.0e-15)
    assert_near(c2s_td[3, 4], -sqrt(3.0) / 2.0, 1.0e-15)
    # every other entry of the d block is zero
    c2s_tnz = 0.0
    for c2s_ti to shape(c2s_td)[0] {
        for c2s_tj to shape(c2s_td)[1] {
            if abs(c2s_td[c2s_ti, c2s_tj]) > 1.0e-15 {
                c2s_tnz = c2s_tnz + 1
            }
        }
    }
    assert_near(c2s_tnz, 8.0, 0.0)
}

test "the spherical functions are orthonormal: M^T G M = I up to l = 6" {
    for c2s_tl to C2S_CLOSED_LMAX + 1 {
        assert_lt(c2s_orth_gap(c2s_closed(c2s_tl), c2s_tl), 1.0e-15)
    }
    c2s_tlorth = 6
    for c2s_tl to c2s_tlorth + 1 {
        assert_lt(c2s_orth_gap(c2s_general(c2s_tl), c2s_tl), 1.0e-14)
    }
}

test "the unit Gram matrix has a unit diagonal and xx overlaps yy by 1/3" {
    c2s_tg = c2s_unit_gram(2)
    for c2s_ti to len(c2s_tg) {
        assert_near(c2s_tg[c2s_ti, c2s_ti], 1.0, 1.0e-15)
    }
    assert_near(c2s_tg[0, 3], 1.0 / 3.0, 2.0e-16)
    assert_near(c2s_tg[0, 1], 0.0, 0.0)
    # the Cartesian row index agrees with bas_powers for every component to l = 5
    c2s_tlrow = 5
    for c2s_tl to c2s_tlrow + 1 {
        c2s_tpw = bas_powers(c2s_tl)
        for c2s_ti to bas_ncart(c2s_tl) {
            assert_near(c2s_cart_row(c2s_tl, c2s_tpw[c2s_ti, 0], c2s_tpw[c2s_ti, 1]), c2s_ti, 0.0)
        }
    }
}

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

from hf.basis import (
    bas_nao_cart, bas_nao_sph, bas_nc, bas_ncart, BAS_NDIM, bas_nsph, bas_odfact, bas_powers,
    shell_cart0, shell_l, shell_sph0,
)

# c2s_closed writes M_l out by hand up to f; above it defers to c2s_general
C2S_CLOSED_LMAX = 3

#| n choose k, exact in doubles for the small arguments used here.
def c2s_binom(n, k) {
    if k < 0 or k > n {
        return 0.0
    }
    r = 1.0
    for i to k {
        r = r * (n - i) / (i + 1)
    }
    return round(r)
}

#| The overlap of the unit Cartesian components of one shell, `[ncart, ncart]`,
#| when they share one radial part. Per axis the Gaussian moment gives
#| `(p + q - 1)!!` for powers p and q with p + q even, and 0 when it is odd;
#| the unit factors `Nc` scale the result so the diagonal is 1.
def c2s_unit_gram(l) -> Arr {
    P = bas_powers(l)
    nc = bas_ncart(l)
    G = zeros(nc, nc)
    #: unit-component overlap: zero for an odd power on any axis, else (2k - 1)!!
    for i to nc {
        for j to nc {
            v = 1.0
            for d to BAS_NDIM {
                s = P[i, d] + P[j, d]
                if mod(s, 2.0) > 0.5 {
                    v = 0.0
                }
                else {
                    v = v * bas_odfact(s / 2)
                }
            }
            G[i, j] = v * bas_nc(P[i, 0], P[i, 1], P[i, 2]) * bas_nc(P[j, 0], P[j, 1], P[j, 2])
        }
    }
    return G
}

#| The row of component (lx, ly, lz) in PySCF's Cartesian order: the
#| components with a larger lx come first, and within one lx a larger ly.
def c2s_cart_row(l, lx, ly) {
    a = l - lx
    return a * (a + 1) / 2 + (a - ly)
}

#| M_l by hand for l <= 3. Rows are the unit Cartesian components in PySCF's
#| order, columns the spherical functions in PySCF's order. Each entry is a
#| coefficient of the polynomial named above it times
#| sqrt((2lx-1)!! (2ly-1)!! (2lz-1)!!), divided by the polynomial's norm.
#| Above l = 3 it returns `c2s_general(l)`.
def c2s_closed(l) -> Arr {
    if l > C2S_CLOSED_LMAX {
        return c2s_general(l)
    }
    M = zeros(bas_ncart(l), bas_nsph(l))
    if l == 0 {
        M[0, 0] = 1.0
    }
    if l == 1 {
        # x, y, z map to themselves
        M[0, 0] = 1.0
        M[1, 1] = 1.0
        M[2, 2] = 1.0
    }
    if l == 2 {
        # rows xx xy xz yy yz zz; columns xy yz z2 xz x2-y2
        # xy, yz, xz: one component each
        M[1, 0] = 1.0
        M[4, 1] = 1.0
        M[2, 3] = 1.0
        # (2z^2 - x^2 - y^2)/2 is sqrt(3) (zz - xx/2 - yy/2) in unit components,
        # and zz - xx/2 - yy/2 has squared norm 3/2 - 2/3 + 1/6 = 1 because
        # two different unit squares overlap by 1/3
        M[5, 2] = 1.0
        M[0, 2] = -0.5
        M[3, 2] = -0.5
        # x^2 - y^2: unit coefficients +-sqrt(3), squared norm 3 + 3 - 2 = 4
        M[0, 4] = sqrt(3.0) / 2.0
        M[3, 4] = -sqrt(3.0) / 2.0
    }
    if l == 3 {
        # rows xxx xxy xxz xyy xyz xzz yyy yyz yzz zzz; columns m = -3 .. 3
        # m = -3, y(3x^2 - y^2)
        M[1, 0] = 3.0 / (2.0 * sqrt(2.0))
        M[6, 0] = -sqrt(10.0) / 4.0
        # m = -2, xyz
        M[4, 1] = 1.0
        # m = -1, y(4z^2 - x^2 - y^2)
        M[8, 2] = sqrt(6.0 / 5.0)
        M[1, 2] = -sqrt(3.0 / 40.0)
        M[6, 2] = -sqrt(6.0) / 4.0
        # m = 0, z(2z^2 - 3x^2 - 3y^2)
        M[9, 3] = 1.0
        M[2, 3] = -3.0 / (2.0 * sqrt(5.0))
        M[7, 3] = -3.0 / (2.0 * sqrt(5.0))
        # m = 1, x(4z^2 - x^2 - y^2)
        M[5, 4] = sqrt(6.0 / 5.0)
        M[0, 4] = -sqrt(6.0) / 4.0
        M[3, 4] = -sqrt(3.0 / 40.0)
        # m = 2, z(x^2 - y^2)
        M[2, 5] = sqrt(3.0) / 2.0
        M[7, 5] = -sqrt(3.0) / 2.0
        # m = 3, x(x^2 - 3y^2)
        M[0, 6] = sqrt(10.0) / 4.0
        M[3, 6] = -3.0 / (2.0 * sqrt(2.0))
    }
    return M
}

#| The column of the spherical function with index m in PySCF's order. For p
#| PySCF keeps x, y, z, which are m = 1, -1, 0.
def c2s_sph_col(l, m) {
    if l == 1 {
        if m == 1 {
            return 0
        }
        if m == -1 {
            return 1
        }
        return 2
    }
    return m + l
}

#| M_l for any l from the real solid harmonic expansion
#|
#|     S_lm ~ sum_t sum_u sum_w (-1)^(t + (w - wm)/2) 4^-t C(l,t) C(l-t,|m|+t) C(t,u) C(|m|,w)
#|            x^(2t + |m| - 2u - w) y^(2u + w) z^(l - 2t - |m|)
#|
#| with t = 0 .. (l - |m|)/2, u = 0 .. t, and w running over the integers up
#| to |m| that are even for m >= 0 and odd for m < 0 (wm is 0 or 1 to
#| match). This is Helgaker's sum with v = w/2. The overall constant is left
#| out: each column is scaled afterwards to unit norm in the metric of
#| `c2s_unit_gram`, which also converts monomial coefficients to unit ones.
def c2s_general(l) -> Arr {
    nc = bas_ncart(l)
    ns = bas_nsph(l)
    A = zeros(nc, ns)
    #: solid harmonic's monomial coefficients, moved to unit components
    for mi to ns {
        m = mi - l
        am = abs(m)
        wm = 0
        if m < 0 {
            wm = 1
        }
        col = c2s_sph_col(l, m)
        for t to floor((l - am) / 2) + 1 {
            for u to t + 1 {
                w = wm
                while w <= am {
                    sgn = 1.0
                    if mod(t + (w - wm) / 2, 2.0) > 0.5 {
                        sgn = -1.0
                    }
                    c = sgn * pow(0.25, t) * c2s_binom(l, t) * c2s_binom(l - t, am + t) * c2s_binom(t, u) * c2s_binom(am, w)
                    lx = 2 * t + am - 2 * u - w
                    ly = 2 * u + w
                    row = c2s_cart_row(l, lx, ly)
                    # a monomial coefficient becomes a unit-component one by
                    # dividing by that component's Nc
                    A[row, col] = A[row, col] + c / bas_nc(lx, ly, l - lx - ly)
                    w = w + 2
                }
            }
        }
    }
    G = c2s_unit_gram(l)
    for j to ns {
        # the column's squared norm in the unit metric, a^T G a, entry by entry
        v = A[:, j]
        W = zeros(nc, nc)
        for a to nc {
            W[a, :] = v[a] * G[a, :] * v
        }
        A[:, j] = v * (1.0 / sqrt(sum(W)))
    }
    return A
}

#| The production M_l: the hand-written table where it exists, the general
#| construction above it.
def c2s_block(l) -> Arr {
    if l <= C2S_CLOSED_LMAX {

return c2s_closed(l)
    }
    return c2s_general(l)
}

#| The block-diagonal M for a basis, `[nao_cart, nao_sph]`: shell s puts its
#| M_l at rows SH_CART0 and columns SH_SPH0, and everything between shells is
#| zero. Then PySCF's spherical X is `M^T X_unit M`.
def c2s_basis(shells: Arr) -> Arr {
    M = zeros(bas_nao_cart(shells), bas_nao_sph(shells))
    for s to len(shells) {
        l = shell_l(shells, s)
        B = c2s_block(l)
        c0 = shell_cart0(shells, s)
        s0 = shell_sph0(shells, s)
        M[c0:c0 + bas_ncart(l), s0:s0 + bas_nsph(l)] = B
    }
    return M
}

# ---- test helpers ----

#| The largest entry of abs(M^T G M - I) for one block.
def c2s_orth_gap(M: Arr, l) {
    E = transpose(M) @ c2s_unit_gram(l) @ M
    for i to len(E) {
        E[i, i] = E[i, i] - 1.0
    }
    return max(abs(E))
}

test "p is x, y, z and d is xy, yz, z2, xz, x2-y2" {
    # p: the identity, so column 0 is x, 1 is y, 2 is z
    c2s_tp = c2s_general(1)
    for c2s_ti to shape(c2s_tp)[0] {
        for c2s_tj to shape(c2s_tp)[1] {
            c2s_tw = 0.0
            if c2s_ti == c2s_tj {
                c2s_tw = 1.0
            }
            assert_near(c2s_tp[c2s_ti, c2s_tj], c2s_tw, 1.0e-15)
        }
    }
    # d rows: xx 0, xy 1, xz 2, yy 3, yz 4, zz 5
    c2s_td = c2s_general(2)
    # column 0 is xy alone, 1 is yz alone, 3 is xz alone
    assert_near(c2s_td[1, 0], 1.0, 1.0e-15)
    assert_near(c2s_td[4, 1], 1.0, 1.0e-15)
    assert_near(c2s_td[2, 3], 1.0, 1.0e-15)
    # column 2 is z^2 - (x^2 + y^2)/2, column 4 is x^2 - y^2
    assert_near(c2s_td[5, 2], 1.0, 1.0e-15)
    assert_near(c2s_td[0, 2], -0.5, 1.0e-15)
    assert_near(c2s_td[3, 2], -0.5, 1.0e-15)
    assert_near(c2s_td[0, 4], sqrt(3.0) / 2.0, 1.0e-15)
    assert_near(c2s_td[3, 4], -sqrt(3.0) / 2.0, 1.0e-15)
    # every other entry of the d block is zero
    c2s_tnz = 0.0
    for c2s_ti to shape(c2s_td)[0] {
        for c2s_tj to shape(c2s_td)[1] {
            if abs(c2s_td[c2s_ti, c2s_tj]) > 1.0e-15 {
                c2s_tnz = c2s_tnz + 1
            }
        }
    }
    assert_near(c2s_tnz, 8.0, 0.0)
}

test "the spherical functions are orthonormal: M^T G M = I up to l = 6" {
    for c2s_tl to C2S_CLOSED_LMAX + 1 {
        assert_lt(c2s_orth_gap(c2s_closed(c2s_tl), c2s_tl), 1.0e-15)
    }
    c2s_tlorth = 6
    for c2s_tl to c2s_tlorth + 1 {
        assert_lt(c2s_orth_gap(c2s_general(c2s_tl), c2s_tl), 1.0e-14)
    }
}

test "the unit Gram matrix has a unit diagonal and xx overlaps yy by 1/3" {
    c2s_tg = c2s_unit_gram(2)
    for c2s_ti to len(c2s_tg) {
        assert_near(c2s_tg[c2s_ti, c2s_ti], 1.0, 1.0e-15)
    }
    assert_near(c2s_tg[0, 3], 1.0 / 3.0, 2.0e-16)
    assert_near(c2s_tg[0, 1], 0.0, 0.0)
    # the Cartesian row index agrees with bas_powers for every component to l = 5
    c2s_tlrow = 5
    for c2s_tl to c2s_tlrow + 1 {
        c2s_tpw = bas_powers(c2s_tl)
        for c2s_ti to bas_ncart(c2s_tl) {
            assert_near(c2s_cart_row(c2s_tl, c2s_tpw[c2s_ti, 0], c2s_tpw[c2s_ti, 1]), c2s_ti, 0.0)
        }
    }
}

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