Notebooks · lib/hf/density

Densities, orbital spaces and the SCF error

The SCF state is a density matrix. For a closed shell it is the total density D = 2 C_occ C_occ^T, the convention PySCF's make_rdm1 uses; for UHF and ROHF it is one density per spin, Da = Ca_occ Ca_occ^T and Db = Cb_occ Cb_occ^T. The J/K stage reads them stacked as one array D[ndens, n, n]; the kernels below that work on one matrix take and return [n, n] (or [m, m] in the orthonormal basis).

The orbitals come from the eigenvectors V of F' = X^T F X, so an occupied orbital is c_k = X V[:, k] and

D = w sum_(k occupied) (X V[:, k]) (X V[:, k])^T,       w = 2 or 1.

Every kernel returns a fresh array. A loop that calls them each iteration runs in the memory of one iteration: a compiled loop frees what an iteration allocated and no variable holds.

Checks a converged density must pass, with N electrons in it and w its occupation (2 for a closed-shell total density, 1 for a spin density):

Tr(D S) = N: the electron count.
D S D = w D: D S / w is a projector, which says the density comes from orthonormal occupied orbitals.
F D S - S D F = 0: the density commutes with the Fock matrix (in the S metric), which is the SCF stationarity condition. den_commutator forms the same commutator in the orthonormal basis, F' P' - P' F' with P' = X^T S D S X, which is what DIIS extrapolates on.

The energy of a density and the Fock matrices built from it is

E = E_nuc + (1/2) sum_d Tr(D_d (H + F_d))

for both conventions: with the total density and F = H + J - K/2 it is the RHF energy, and with spin densities and F_s = H + J - K_s it is the UHF energy.

<S^2> of a UHF determinant is

<S^2> = ((Na - Nb)/2)^2 + (Na + Nb)/2 - Tr(Da S Db S),

where the trace counts how far the alpha and beta orbitals are from being the same; it is Nb (and <S^2> = S(S+1)) when every beta orbital is also an alpha one.

Used with import hf.density.

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[1]lines 51–276

import hf.opts
import hf.eigen

#| A `[1, n, n]` stack holding one matrix.
def den_stack1(A: Arr) {
    n = len(A)
    D = zeros(1, n, n)
    setslab(D, 0, A)
    return D
}

#| A `[2, n, n]` stack holding two matrices.
def den_stack2(A: Arr, B: Arr) {
    n = len(A)
    D = zeros(2, n, n)
    setslab(D, 0, A)
    setslab(D, 1, B)
    return D
}

#| `w sum_(k = k0 .. k1-1) (X V[:, k]) (X V[:, k])^T`, `[n, n]`, with
#| `X[n, m]` (its first m columns) and the eigenvectors `V[m, m]` as columns.
def den_from_vectors(X: Arr, V: Arr, m, k0, k1, w) {
    n = len(X)
    if k1 <= k0 {
        return zeros(n, n)
    }
    C = X[:, 0:m] @ V[0:m, k0:k1]
    return (w * C) @ transpose(C)
}

#| The Fock matrices from J and K, `[nd, n, n]`: `H + J - K/2` for one total
#| density, `H + J_a + J_b - K_s` for two spin densities.
def den_fock(H: Arr, JK: Arr, nd) {
    n = len(H)
    F = zeros(nd, n, n)
    if nd < 2 {
        setslab(F, 0, H + slab(JK, 0) - 0.5 * slab(JK, 1))
    }
    else {
        jt = slab(JK, 0) + slab(JK, 2)
        setslab(F, 0, H + jt - slab(JK, 1))
        setslab(F, 1, H + jt - slab(JK, 3))
    }
    return F
}

#| `X^T A X` over the first m columns of X: a matrix in the orthonormal basis.
def den_orth(X: Arr, A: Arr, m) {
    if m == shape(X)[1] {
        return transpose(X) @ (A @ X)
    }
    Xm = X[:, 0:m]
    return transpose(Xm) @ (A @ Xm)
}

#| `X^T S D S X`: the density D in the orthonormal basis.
def den_orth_density(X: Arr, S: Arr, D: Arr, m) {
    if m == shape(X)[1] {
        return den_orth_density_sx(S @ X, D)
    }
    return den_orth_density_sx(S @ X[:, 0:m], D)
}

#| `U^T D U` with `U = S X` computed once: the density D in the orthonormal
#| basis, for a loop that moves a new density there every iteration.
def den_orth_density_sx(U: Arr, D: Arr) {
    return transpose(U) @ (D @ U)
}

#| `F' P' - P' F'` for two `[m, m]` matrices: the SCF error in the
#| orthonormal basis.
def den_commutator(Fp: Arr, Pp: Arr) {
    m = len(Fp)
    E = zeros(m, m)
    for a to m {
        for b to m {
            s = 0.0
            for c to m {
                s = s + Fp[a, c] * Pp[c, b] - Pp[a, c] * Fp[c, b]
            }
            E[a, b] = s
        }
    }
    return E
}

#| `F' P' - P' F'` for two symmetric `[m, m]` matrices, as `A - A^T` with
#| `A = F' P'`: one matrix product instead of two, since `(F' P')^T = P' F'`
#| when both are symmetric. The result is antisymmetric to the bit.
#| `den_commutator` is the definition term by term, and the test below
#| holds the two together.
def den_commutator_sym(Fp: Arr, Pp: Arr) {
    A = Fp @ Pp
    return A - transpose(A)
}

#| The largest |entry| of a matrix.
def den_maxabs(A: Arr) {
    return max(abs(A))
}

#| max |F D S - S D F| in the AO basis, for density d and Fock matrix f.
def den_ao_commutator(F: Arr, f, D: Arr, d, S: Arr) {
    # S D F = (F D S)^T for symmetric F, D, S
    FDS = slab(F, f) @ slab(D, d) @ S
    return max(abs(FDS - transpose(FDS)))
}

#| Tr(D[d] S).
def den_trace(D: Arr, d, S: Arr) {
    return sum(slab(D, d) * transpose(S))
}

#| Tr(D[a] S D[b] S).
def den_trace2(D: Arr, a, b, S: Arr) {
    P = slab(D, a) @ S
    Q = slab(D, b) @ S
    return sum(P * transpose(Q))
}

#| max |D S D - w D| for density d.
def den_idempotency(D: Arr, d, S: Arr, w) {
    A = slab(D, d)
    return max(abs(A @ S @ A - w * A))
}

#| Frobenius norm of D[d].
def den_fnorm(D: Arr, d) {
    A = slab(D, d)
    return sqrt(sum(A * A))
}

#| max |A[a] - B[b]| over two stacks of matrices.
def den_gap(A: Arr, a, B: Arr, b) {
    return max(abs(slab(A, a) - slab(B, b)))
}

#| `E_nuc + (1/2) sum_d Tr(D_d (H + F_d))` over `ndens` densities.
def den_energy(H: Arr, F: Arr, D: Arr, ndens, enuc) {
    n = len(H)
    Hs = zeros(ndens, n, n)
    for d to ndens {
        setslab(Hs, d, H)
    }
    return enuc + 0.5 * sum(D * (Hs + F))
}

#| `<S^2>` of a UHF pair of spin densities D[0] (alpha), D[1] (beta).
def den_s2(D: Arr, S: Arr, na, nb) {
    sz = 0.5 * (na - nb)
    return sz * sz + 0.5 * (na + nb) - den_trace2(D, 0, 1, S)
}

#| Indices of the `k` smallest entries of `v`, ascending by value; entries
#| whose `mask` is `1` are skipped. Ties go to the lower index.
def den_lowest(v: Arr, k, mask: Arr) {
    m = len(v)
    taken = zeros(m)
    idx = []
    for c to k {
        best = -1
        for a to m {
            if mask[a] < 0.5 and taken[a] < 0.5 {
                if best < 0 {
                    best = a
                }
                else if v[a] < v[best] {
                    best = a
                }
            }
        }
        taken[best] = 1
        idx = append(idx, best)
    }
    return idx
}

test "a density built from orbitals is the projector it should be" {
    # with X = I and V = I the density is w times the first k unit vectors
    den_X = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]
    den_n = len(den_X)
    den_D = den_stack1(den_from_vectors(den_X, den_X, den_n, 0, 2, 2.0))
    assert_near(den_D[0, 0, 0] + den_D[0, 1, 1] + den_D[0, 2, 2], 4.0, 0.0)
    assert_near(den_D[0, 2, 2], 0.0, 0.0)
    assert_near(den_idempotency(den_D, 0, den_X, 2.0), 0.0, 0.0)
    # the orthonormal-basis density of an AO density is back to diagonal form
    den_P = den_stack1(den_orth_density(den_X, den_X, slab(den_D, 0), den_n))
    assert_near(den_gap(den_P, 0, den_D, 0), 0.0, 0.0)
    # and a density commutes with itself
    assert_near(den_maxabs(den_commutator(slab(den_P, 0), slab(den_P, 0))), 0.0, 0.0)
}

test "the one-product commutator is the term-by-term one on symmetric matrices" {
    den_m = 7
    den_F = zeros(den_m, den_m)
    den_Q = zeros(den_m, den_m)
    for den_a to den_m {
        for den_b to den_m {
            den_F[den_a, den_b] = sin(1.0 + den_a + den_b) + 0.1 * (den_a * den_b)
            den_Q[den_a, den_b] = cos(0.5 * den_a * den_b) - 0.3 * (den_a + den_b)
        }
    }
    den_E0 = den_commutator(den_F, den_Q)
    den_E1 = den_commutator_sym(den_F, den_Q)
    # same matrix to rounding, relative to its size; exactly antisymmetric
    assert_lt(max(abs(den_E1 - den_E0)), 1.0e-14 * max(abs(den_E0)))
    assert_lt(1.0, max(abs(den_E0)))
    assert_near(max(abs(den_E1 + transpose(den_E1))), 0.0, 0.0)
}

test "the lowest-k picker skips masked entries and breaks ties low" {
    den_v = [0.5, -1.0, 0.5, -3.0, 2.0]
    den_mask = [0, 0, 0, 1, 0]
    den_k = 3
    den_idx = den_lowest(den_v, den_k, den_mask)
    den_last = den_v[den_idx[den_k - 1]]

    assert_near(den_idx[0], 1, 0.0)
    assert_near(den_idx[1], 0, 0.0)
    assert_near(den_idx[2], 2, 0.0)
    assert_near(den_last, 0.5, 0.0)
}

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

import hf.opts
import hf.eigen

#| A `[1, n, n]` stack holding one matrix.
def den_stack1(A: Arr) {
    n = len(A)
    D = zeros(1, n, n)
    setslab(D, 0, A)
    return D
}

#| A `[2, n, n]` stack holding two matrices.
def den_stack2(A: Arr, B: Arr) {
    n = len(A)
    D = zeros(2, n, n)
    setslab(D, 0, A)
    setslab(D, 1, B)
    return D
}

#| `w sum_(k = k0 .. k1-1) (X V[:, k]) (X V[:, k])^T`, `[n, n]`, with
#| `X[n, m]` (its first m columns) and the eigenvectors `V[m, m]` as columns.
def den_from_vectors(X: Arr, V: Arr, m, k0, k1, w) {
    n = len(X)
    if k1 <= k0 {
        return zeros(n, n)
    }
    C = X[:, 0:m] @ V[0:m, k0:k1]
    return (w * C) @ transpose(C)
}

#| The Fock matrices from J and K, `[nd, n, n]`: `H + J - K/2` for one total
#| density, `H + J_a + J_b - K_s` for two spin densities.
def den_fock(H: Arr, JK: Arr, nd) {
    n = len(H)
    F = zeros(nd, n, n)
    if nd < 2 {
        setslab(F, 0, H + slab(JK, 0) - 0.5 * slab(JK, 1))
    }
    else {
        jt = slab(JK, 0) + slab(JK, 2)
        setslab(F, 0, H + jt - slab(JK, 1))
        setslab(F, 1, H + jt - slab(JK, 3))
    }
    return F
}

#| `X^T A X` over the first m columns of X: a matrix in the orthonormal basis.
def den_orth(X: Arr, A: Arr, m) {
    if m == shape(X)[1] {
        return transpose(X) @ (A @ X)
    }
    Xm = X[:, 0:m]
    return transpose(Xm) @ (A @ Xm)
}

#| `X^T S D S X`: the density D in the orthonormal basis.
def den_orth_density(X: Arr, S: Arr, D: Arr, m) {
    if m == shape(X)[1] {
        return den_orth_density_sx(S @ X, D)
    }
    return den_orth_density_sx(S @ X[:, 0:m], D)
}

#| `U^T D U` with `U = S X` computed once: the density D in the orthonormal
#| basis, for a loop that moves a new density there every iteration.
def den_orth_density_sx(U: Arr, D: Arr) {
    return transpose(U) @ (D @ U)
}

#| `F' P' - P' F'` for two `[m, m]` matrices: the SCF error in the
#| orthonormal basis.
def den_commutator(Fp: Arr, Pp: Arr) {
    m = len(Fp)
    E = zeros(m, m)
    for a to m {
        for b to m {
            s = 0.0
            for c to m {
                s = s + Fp[a, c] * Pp[c, b] - Pp[a, c] * Fp[c, b]
            }
            E[a, b] = s
        }
    }
    return E
}

#| `F' P' - P' F'` for two symmetric `[m, m]` matrices, as `A - A^T` with
#| `A = F' P'`: one matrix product instead of two, since `(F' P')^T = P' F'`
#| when both are symmetric. The result is antisymmetric to the bit.
#| `den_commutator` is the definition term by term, and the test below
#| holds the two together.
def den_commutator_sym(Fp: Arr, Pp: Arr) {
    A = Fp @ Pp
    return A - transpose(A)
}

#| The largest |entry| of a matrix.
def den_maxabs(A: Arr) {
    return max(abs(A))
}

#| max |F D S - S D F| in the AO basis, for density d and Fock matrix f.
def den_ao_commutator(F: Arr, f, D: Arr, d, S: Arr) {
    # S D F = (F D S)^T for symmetric F, D, S
    FDS = slab(F, f) @ slab(D, d) @ S
    return max(abs(FDS - transpose(FDS)))
}

#| Tr(D[d] S).
def den_trace(D: Arr, d, S: Arr) {
    return sum(slab(D, d) * transpose(S))
}

#| Tr(D[a] S D[b] S).
def den_trace2(D: Arr, a, b, S: Arr) {
    P = slab(D, a) @ S
    Q = slab(D, b) @ S
    return sum(P * transpose(Q))
}

#| max |D S D - w D| for density d.
def den_idempotency(D: Arr, d, S: Arr, w) {
    A = slab(D, d)
    return max(abs(A @ S @ A - w * A))
}

#| Frobenius norm of D[d].
def den_fnorm(D: Arr, d) {
    A = slab(D, d)
    return sqrt(sum(A * A))
}

#| max |A[a] - B[b]| over two stacks of matrices.
def den_gap(A: Arr, a, B: Arr, b) {
    return max(abs(slab(A, a) - slab(B, b)))
}

#| `E_nuc + (1/2) sum_d Tr(D_d (H + F_d))` over `ndens` densities.
def den_energy(H: Arr, F: Arr, D: Arr, ndens, enuc) {
    n = len(H)
    Hs = zeros(ndens, n, n)
    for d to ndens {
        setslab(Hs, d, H)
    }
    return enuc + 0.5 * sum(D * (Hs + F))
}

#| `<S^2>` of a UHF pair of spin densities D[0] (alpha), D[1] (beta).
def den_s2(D: Arr, S: Arr, na, nb) {
    sz = 0.5 * (na - nb)
    return sz * sz + 0.5 * (na + nb) - den_trace2(D, 0, 1, S)
}

#| Indices of the `k` smallest entries of `v`, ascending by value; entries
#| whose `mask` is `1` are skipped. Ties go to the lower index.
def den_lowest(v: Arr, k, mask: Arr) {
    m = len(v)
    taken = zeros(m)
    idx = []
    for c to k {
        best = -1
        for a to m {
            if mask[a] < 0.5 and taken[a] < 0.5 {
                if best < 0 {
                    best = a
                }
                else if v[a] < v[best] {
                    best = a
                }
            }
        }
        taken[best] = 1
        idx = append(idx, best)
    }
    return idx
}

test "a density built from orbitals is the projector it should be" {
    # with X = I and V = I the density is w times the first k unit vectors
    den_X = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]
    den_n = len(den_X)
    den_D = den_stack1(den_from_vectors(den_X, den_X, den_n, 0, 2, 2.0))
    assert_near(den_D[0, 0, 0] + den_D[0, 1, 1] + den_D[0, 2, 2], 4.0, 0.0)
    assert_near(den_D[0, 2, 2], 0.0, 0.0)
    assert_near(den_idempotency(den_D, 0, den_X, 2.0), 0.0, 0.0)
    # the orthonormal-basis density of an AO density is back to diagonal form
    den_P = den_stack1(den_orth_density(den_X, den_X, slab(den_D, 0), den_n))
    assert_near(den_gap(den_P, 0, den_D, 0), 0.0, 0.0)
    # and a density commutes with itself
    assert_near(den_maxabs(den_commutator(slab(den_P, 0), slab(den_P, 0))), 0.0, 0.0)
}

test "the one-product commutator is the term-by-term one on symmetric matrices" {
    den_m = 7
    den_F = zeros(den_m, den_m)
    den_Q = zeros(den_m, den_m)
    for den_a to den_m {
        for den_b to den_m {
            den_F[den_a, den_b] = sin(1.0 + den_a + den_b) + 0.1 * (den_a * den_b)
            den_Q[den_a, den_b] = cos(0.5 * den_a * den_b) - 0.3 * (den_a + den_b)
        }
    }
    den_E0 = den_commutator(den_F, den_Q)
    den_E1 = den_commutator_sym(den_F, den_Q)
    # same matrix to rounding, relative to its size; exactly antisymmetric
    assert_lt(max(abs(den_E1 - den_E0)), 1.0e-14 * max(abs(den_E0)))
    assert_lt(1.0, max(abs(den_E0)))
    assert_near(max(abs(den_E1 + transpose(den_E1))), 0.0, 0.0)
}

test "the lowest-k picker skips masked entries and breaks ties low" {
    den_v = [0.5, -1.0, 0.5, -3.0, 2.0]
    den_mask = [0, 0, 0, 1, 0]
    den_k = 3
    den_idx = den_lowest(den_v, den_k, den_mask)
    den_last = den_v[den_idx[den_k - 1]]

assert_near(den_idx[0], 1, 0.0)
    assert_near(den_idx[1], 0, 0.0)
    assert_near(den_idx[2], 2, 0.0)
    assert_near(den_last, 0.5, 0.0)
}

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