# Molecule-driven restricted Hartree-Fock in STO-3G.
#
# The basis is assembled from an element library (H, He, C, N, O, F) given only
# the atoms; the McMurchie-Davidson engine handles the s/p integrals. This is the
# reusable driver behind the molecule database (tests/molecule_db.rs), which
# substitutes the MOL_BEGIN..MOL_END block per molecule and checks the energy
# against known values. Default molecule = water (E = -74.9420787, Crawford).

pi = 3.141592653589793
n_prim = 3

# =====================================================================
# McMurchie-Davidson integral engine (molecule-independent).
# =====================================================================

# Boys function F_n(T) = integral_0^1 t^(2n) exp(-T t^2) dt.
def boys_n(nn, T) {
    if T < 1.0e-12 {
        return 1.0 / (2.0 * nn + 1.0)
    }
    return simpson(fn(t) -> pow(t, 2.0 * nn) * exp(-T * t * t), 0.0, 1.0, 100)
}

# Hermite expansion coefficient E_t^{ij} (1D), exponents a,b, Qx = Ax - Bx.
def hermite_E(i, j, t, Qx, a, b) {
    if i < 0.0 { return 0.0 }
    if j < 0.0 { return 0.0 }
    if t < 0.0 { return 0.0 }
    if t > i + j { return 0.0 }
    p = a + b
    if i == 0.0 {
        if j == 0.0 {
            mu = a * b / p
            return exp(-mu * Qx * Qx)
        }
        xpb = a / p * Qx
        return (0.5 / p * hermite_E(i, j - 1.0, t - 1.0, Qx, a, b)
             + xpb * hermite_E(i, j - 1.0, t, Qx, a, b)
             + (t + 1.0) * hermite_E(i, j - 1.0, t + 1.0, Qx, a, b))
    }
    xpa = -b / p * Qx
    return (0.5 / p * hermite_E(i - 1.0, j, t - 1.0, Qx, a, b)
         + xpa * hermite_E(i - 1.0, j, t, Qx, a, b)
         + (t + 1.0) * hermite_E(i - 1.0, j, t + 1.0, Qx, a, b))
}

# Hermite Coulomb integral R_{tuv}^n.
def hermite_R(t, u, v, nn, p, PCx, PCy, PCz, RPC2) {
    if t == 0.0 {
        if u == 0.0 {
            if v == 0.0 {
                return pow(-2.0 * p, nn) * boys_n(nn, p * RPC2)
            }
            res = PCz * hermite_R(t, u, v - 1.0, nn + 1.0, p, PCx, PCy, PCz, RPC2)
            if v > 1.0 {
                res += (v - 1.0) * hermite_R(t, u, v - 2.0, nn + 1.0, p, PCx, PCy, PCz, RPC2)
            }
            return res
        }
        res = PCy * hermite_R(t, u - 1.0, v, nn + 1.0, p, PCx, PCy, PCz, RPC2)
        if u > 1.0 {
            res += (u - 1.0) * hermite_R(t, u - 2.0, v, nn + 1.0, p, PCx, PCy, PCz, RPC2)
        }
        return res
    }
    res = PCx * hermite_R(t - 1.0, u, v, nn + 1.0, p, PCx, PCy, PCz, RPC2)
    if t > 1.0 {
        res += (t - 1.0) * hermite_R(t - 2.0, u, v, nn + 1.0, p, PCx, PCy, PCz, RPC2)
    }
    return res
}

def overlap_1d(i, j, Qx, a, b) {
    return hermite_E(i, j, 0.0, Qx, a, b) * sqrt(pi / (a + b))
}

# 1D kinetic energy matrix element (differentiate the ket twice).
def kinetic_1d(i, j, Qx, a, b) {
    term = b * (2.0 * j + 1.0) * overlap_1d(i, j, Qx, a, b)
    term += -2.0 * b * b * overlap_1d(i, j + 2.0, Qx, a, b)
    term += -0.5 * j * (j - 1.0) * overlap_1d(i, j - 2.0, Qx, a, b)
    return term
}

# =====================================================================
# Primitive integrals over two/four basis functions (reading bf data
# by index from the globals defined in the MOLECULE/BASIS block).
# =====================================================================

def prim_overlap(a, b, mu, nu) {
    sx = overlap_1d(bfL[mu], bfL[nu], bfCx[mu] - bfCx[nu], a, b)
    sy = overlap_1d(bfM[mu], bfM[nu], bfCy[mu] - bfCy[nu], a, b)
    sz = overlap_1d(bfN[mu], bfN[nu], bfCz[mu] - bfCz[nu], a, b)
    return sx * sy * sz
}

def prim_kinetic(a, b, mu, nu) {
    sx = overlap_1d(bfL[mu], bfL[nu], bfCx[mu] - bfCx[nu], a, b)
    sy = overlap_1d(bfM[mu], bfM[nu], bfCy[mu] - bfCy[nu], a, b)
    sz = overlap_1d(bfN[mu], bfN[nu], bfCz[mu] - bfCz[nu], a, b)
    tx = kinetic_1d(bfL[mu], bfL[nu], bfCx[mu] - bfCx[nu], a, b)
    ty = kinetic_1d(bfM[mu], bfM[nu], bfCy[mu] - bfCy[nu], a, b)
    tz = kinetic_1d(bfN[mu], bfN[nu], bfCz[mu] - bfCz[nu], a, b)
    return tx * sy * sz + sx * ty * sz + sx * sy * tz
}

def prim_nuclear(a, b, mu, nu, Cx, Cy, Cz) {
    la = bfL[mu]
    ma = bfM[mu]
    na = bfN[mu]
    lb = bfL[nu]
    mb = bfM[nu]
    nb = bfN[nu]
    p = a + b
    Px = (a * bfCx[mu] + b * bfCx[nu]) / p
    Py = (a * bfCy[mu] + b * bfCy[nu]) / p
    Pz = (a * bfCz[mu] + b * bfCz[nu]) / p
    PCx = Px - Cx
    PCy = Py - Cy
    PCz = Pz - Cz
    RPC2 = PCx * PCx + PCy * PCy + PCz * PCz
    qx = bfCx[mu] - bfCx[nu]
    qy = bfCy[mu] - bfCy[nu]
    qz = bfCz[mu] - bfCz[nu]
    val = 0.0
    for t to la + lb + 1.0 {
        for u to ma + mb + 1.0 {
            for v to na + nb + 1.0 {
                e = hermite_E(la, lb, t, qx, a, b) * hermite_E(ma, mb, u, qy, a, b) * hermite_E(na, nb, v, qz, a, b)
                val += e * hermite_R(t, u, v, 0.0, p, PCx, PCy, PCz, RPC2)
            }
        }
    }
    return 2.0 * pi / p * val
}

def prim_eri(a, b, c, d, mu, nu, lam, sig) {
    p = a + b
    q = c + d
    Px = (a * bfCx[mu] + b * bfCx[nu]) / p
    Py = (a * bfCy[mu] + b * bfCy[nu]) / p
    Pz = (a * bfCz[mu] + b * bfCz[nu]) / p
    Qx = (c * bfCx[lam] + d * bfCx[sig]) / q
    Qy = (c * bfCy[lam] + d * bfCy[sig]) / q
    Qz = (c * bfCz[lam] + d * bfCz[sig]) / q
    alpha = p * q / (p + q)
    PQx = Px - Qx
    PQy = Py - Qy
    PQz = Pz - Qz
    RPQ2 = PQx * PQx + PQy * PQy + PQz * PQz
    qx1 = bfCx[mu] - bfCx[nu]
    qy1 = bfCy[mu] - bfCy[nu]
    qz1 = bfCz[mu] - bfCz[nu]
    qx2 = bfCx[lam] - bfCx[sig]
    qy2 = bfCy[lam] - bfCy[sig]
    qz2 = bfCz[lam] - bfCz[sig]
    val = 0.0
    for t1 to bfL[mu] + bfL[nu] + 1.0 {
        for u1 to bfM[mu] + bfM[nu] + 1.0 {
            for v1 to bfN[mu] + bfN[nu] + 1.0 {
                e1 = hermite_E(bfL[mu], bfL[nu], t1, qx1, a, b) * hermite_E(bfM[mu], bfM[nu], u1, qy1, a, b) * hermite_E(bfN[mu], bfN[nu], v1, qz1, a, b)
                for t2 to bfL[lam] + bfL[sig] + 1.0 {
                    for u2 to bfM[lam] + bfM[sig] + 1.0 {
                        for v2 to bfN[lam] + bfN[sig] + 1.0 {
                            e2 = hermite_E(bfL[lam], bfL[sig], t2, qx2, c, d) * hermite_E(bfM[lam], bfM[sig], u2, qy2, c, d) * hermite_E(bfN[lam], bfN[sig], v2, qz2, c, d)
                            sign = pow(-1.0, t2 + u2 + v2)
                            val += e1 * e2 * sign * hermite_R(t1 + t2, u1 + u2, v1 + v2, 0.0, alpha, PQx, PQy, PQz, RPQ2)
                        }
                    }
                }
            }
        }
    }
    return 2.0 * pow(pi, 2.5) / (p * q * sqrt(p + q)) * val
}

# Contracted integrals: sum primitive integrals weighted by contraction coeffs.
def cont_overlap(mu, nu) {
    s = 0.0
    for ka to n_prim {
        for kb to n_prim {
            s += bfC[mu, ka] * bfC[nu, kb] * prim_overlap(bfA[mu, ka], bfA[nu, kb], mu, nu)
        }
    }
    return s
}

def cont_kinetic(mu, nu) {
    s = 0.0
    for ka to n_prim {
        for kb to n_prim {
            s += bfC[mu, ka] * bfC[nu, kb] * prim_kinetic(bfA[mu, ka], bfA[nu, kb], mu, nu)
        }
    }
    return s
}

def cont_nuclear(mu, nu, Cx, Cy, Cz) {
    s = 0.0
    for ka to n_prim {
        for kb to n_prim {
            s += bfC[mu, ka] * bfC[nu, kb] * prim_nuclear(bfA[mu, ka], bfA[nu, kb], mu, nu, Cx, Cy, Cz)
        }
    }
    return s
}

def cont_eri(mu, nu, lam, sig) {
    s = 0.0
    for ka to n_prim {
        for kb to n_prim {
            for kc to n_prim {
                for kd to n_prim {
                    coef = bfC[mu, ka] * bfC[nu, kb] * bfC[lam, kc] * bfC[sig, kd]
                    s += coef * prim_eri(bfA[mu, ka], bfA[nu, kb], bfA[lam, kc], bfA[sig, kd], mu, nu, lam, sig)
                }
            }
        }
    }
    return s
}

# =====================================================================
# MOLECULE-DRIVEN STO-3G HARTREE-FOCK.
# The basis is built automatically from an element library given only the
# atoms (nuclear charge + position, in bohr). The molecule block below is the
# default (water, Crawford geometry, E = -74.9420787); the database harness
# substitutes the MOL_BEGIN..MOL_END section for each molecule.
# =====================================================================

# === MOL_BEGIN ===
n_atoms = 3
mol_charge = 0.0
atom_Z = zeros(n_atoms)
atom_x = zeros(n_atoms)
atom_y = zeros(n_atoms)
atom_z = zeros(n_atoms)
atom_Z[0] = 8.0
atom_x[0] = 0.0
atom_y[0] = -0.143225816552
atom_z[0] = 0.0
atom_Z[1] = 1.0
atom_x[1] = 1.638036840407
atom_y[1] = 1.136548822547
atom_z[1] = 0.0
atom_Z[2] = 1.0
atom_x[2] = -1.638036840407
atom_y[2] = 1.136548822547
atom_z[2] = 0.0
# === MOL_END ===

# STO-3G primitive exponents per element (rows indexed by nuclear charge Z).
# The contraction coefficients are element-independent (canonical STO-3G).
# e1s = 1s exponents; esp = 2sp (shared 2s/2p); e3sp = 3sp (shared 3s/3p, period 3).
e1s = zeros(19, 3)
esp = zeros(19, 3)
e3sp = zeros(19, 3)
e1s[1, 0] = 3.42525091
e1s[1, 1] = 0.62391373
e1s[1, 2] = 0.16885540
e1s[2, 0] = 6.36242139
e1s[2, 1] = 1.15892300
e1s[2, 2] = 0.31364979
e1s[3, 0] = 16.1195750
e1s[3, 1] = 2.9362007
e1s[3, 2] = 0.7946505
esp[3, 0] = 0.6362897
esp[3, 1] = 0.1478601
esp[3, 2] = 0.0480887
e1s[4, 0] = 30.1678710
e1s[4, 1] = 5.4951153
e1s[4, 2] = 1.4871927
esp[4, 0] = 1.3148331
esp[4, 1] = 0.3055389
esp[4, 2] = 0.0993707
e1s[5, 0] = 48.7911130
e1s[5, 1] = 8.8873622
e1s[5, 2] = 2.4052670
esp[5, 0] = 2.2369561
esp[5, 1] = 0.5198205
esp[5, 2] = 0.1690618
e1s[6, 0] = 71.6168370
e1s[6, 1] = 13.0450960
e1s[6, 2] = 3.5305122
esp[6, 0] = 2.9412494
esp[6, 1] = 0.6834831
esp[6, 2] = 0.2222899
e1s[7, 0] = 99.1061690
e1s[7, 1] = 18.0523120
e1s[7, 2] = 4.8856602
esp[7, 0] = 3.7804559
esp[7, 1] = 0.8784966
esp[7, 2] = 0.2857144
e1s[8, 0] = 130.7093200
e1s[8, 1] = 23.8088610
e1s[8, 2] = 6.4436083
esp[8, 0] = 5.0331513
esp[8, 1] = 1.1695961
esp[8, 2] = 0.3803890
e1s[9, 0] = 166.6791300
e1s[9, 1] = 30.3608120
e1s[9, 2] = 8.2168207
esp[9, 0] = 6.4648032
esp[9, 1] = 1.5022812
esp[9, 2] = 0.4885885
e1s[10, 0] = 207.0156100
e1s[10, 1] = 37.7081510
e1s[10, 2] = 10.2052970
esp[10, 0] = 8.2463151
esp[10, 1] = 1.9162662
esp[10, 2] = 0.6232293
# Period 3 (Na-Ar) adds a 3s/3p shell (e3sp), with element-independent 3s/3p
# contraction coefficients (in the builder below). Only Mg is populated here, as a
# validation probe of the period-3 machinery: the Mg atom gives -197.02, matching
# the known closed-shell STO-3G value. The remaining period-3 exponents must be
# copied from a basis-set source (e.g. Basis Set Exchange) before use -- this is
# deliberately left blank rather than reconstructed from memory.
e1s[12, 0] = 299.2374000
e1s[12, 1] = 54.5064700
e1s[12, 2] = 14.7515750
esp[12, 0] = 15.1218200
esp[12, 1] = 3.5135930
esp[12, 2] = 1.1428570
e3sp[12, 0] = 1.3954482
e3sp[12, 1] = 0.3893265
e3sp[12, 2] = 0.1523267

# Count basis functions and electrons. H/He: 1s only (1 function); period 2
# (Li-Ne): 1s,2s,2p (5); period 3 (Na-Ar): +3s,3p (9).
n_basis = 0.0
n_elec = 0.0
for i to n_atoms {
    z = atom_Z[i]
    n_elec += z
    if z <= 2.0 {
        n_basis += 1.0
    }
    else {
        if z <= 10.0 {
            n_basis += 5.0
        }
        else {
            n_basis += 9.0
        }
    }
}
n_elec -= mol_charge
n_occ = n_elec / 2.0

bfL = zeros(n_basis)
bfM = zeros(n_basis)
bfN = zeros(n_basis)
bfCx = zeros(n_basis)
bfCy = zeros(n_basis)
bfCz = zeros(n_basis)
bfA = zeros(n_basis, n_prim)
bfC = zeros(n_basis, n_prim)

# Set basis function `idx`: center, angular momentum (l,m,n), and three
# primitive exponents / (normalized-primitive) contraction coefficients.
def set_bf(idx, cx, cy, cz, l, m, n, a0, a1, a2, c0, c1, c2) {
    bfCx[idx] = cx
    bfCy[idx] = cy
    bfCz[idx] = cz
    bfL[idx] = l
    bfM[idx] = m
    bfN[idx] = n
    bfA[idx, 0] = a0
    bfA[idx, 1] = a1
    bfA[idx, 2] = a2
    lt = l + m + n
    bfC[idx, 0] = c0 * pow(2.0 * a0 / pi, 0.75) * pow(4.0 * a0, lt / 2.0)
    bfC[idx, 1] = c1 * pow(2.0 * a1 / pi, 0.75) * pow(4.0 * a1, lt / 2.0)
    bfC[idx, 2] = c2 * pow(2.0 * a2 / pi, 0.75) * pow(4.0 * a2, lt / 2.0)
}

# Build the basis: a 1s on every atom, plus 2s and 2p (x,y,z) on first-row atoms.
idx = 0.0
for i to n_atoms {
    z = atom_Z[i]
    cx = atom_x[i]
    cy = atom_y[i]
    cz = atom_z[i]
    set_bf(idx, cx, cy, cz, 0.0, 0.0, 0.0, e1s[z, 0], e1s[z, 1], e1s[z, 2], 0.15432897, 0.53532814, 0.44463454)
    idx += 1.0
    if z > 2.0 {
        set_bf(idx, cx, cy, cz, 0.0, 0.0, 0.0, esp[z, 0], esp[z, 1], esp[z, 2], -0.09996723, 0.39951283, 0.70011547)
        idx += 1.0
        set_bf(idx, cx, cy, cz, 1.0, 0.0, 0.0, esp[z, 0], esp[z, 1], esp[z, 2], 0.15591627, 0.60768372, 0.39195739)
        idx += 1.0
        set_bf(idx, cx, cy, cz, 0.0, 1.0, 0.0, esp[z, 0], esp[z, 1], esp[z, 2], 0.15591627, 0.60768372, 0.39195739)
        idx += 1.0
        set_bf(idx, cx, cy, cz, 0.0, 0.0, 1.0, esp[z, 0], esp[z, 1], esp[z, 2], 0.15591627, 0.60768372, 0.39195739)
        idx += 1.0
        if z > 10.0 {
            set_bf(idx, cx, cy, cz, 0.0, 0.0, 0.0, e3sp[z, 0], e3sp[z, 1], e3sp[z, 2], -0.3088442, 0.1960641, 0.8542530)
            idx += 1.0
            set_bf(idx, cx, cy, cz, 1.0, 0.0, 0.0, e3sp[z, 0], e3sp[z, 1], e3sp[z, 2], -0.1215468, 0.5715228, 0.5498949)
            idx += 1.0
            set_bf(idx, cx, cy, cz, 0.0, 1.0, 0.0, e3sp[z, 0], e3sp[z, 1], e3sp[z, 2], -0.1215468, 0.5715228, 0.5498949)
            idx += 1.0
            set_bf(idx, cx, cy, cz, 0.0, 0.0, 1.0, e3sp[z, 0], e3sp[z, 1], e3sp[z, 2], -0.1215468, 0.5715228, 0.5498949)
            idx += 1.0
        }
    }
}

# Normalize each contracted basis function so <mu|mu> = 1.
for i to n_basis {
    nrm = 1.0 / sqrt(cont_overlap(i, i))
    for k to n_prim {
        bfC[i, k] = bfC[i, k] * nrm
    }
}

# =====================================================================
# Build integrals and run the SCF.
# =====================================================================
S = zeros(n_basis, n_basis)
H_core = zeros(n_basis, n_basis)
for mu to n_basis {
    for nu to n_basis {
        S[mu, nu] = cont_overlap(mu, nu)
        v = 0.0
        for atom to n_atoms {
            v += -atom_Z[atom] * cont_nuclear(mu, nu, atom_x[atom], atom_y[atom], atom_z[atom])
        }
        H_core[mu, nu] = cont_kinetic(mu, nu) + v
    }
}

# Two-electron integrals, exploiting the 8-fold permutational symmetry
# (mu nu|lam sig) = (nu mu|lam sig) = (mu nu|sig lam) = (lam sig|mu nu) = ...
# Pass 1 (parallel, the expensive part): evaluate only the unique/canonical
# integrals -- ~8x fewer cont_eri calls. A tuple is canonical when mu>=nu,
# lam>=sig, and idx(mu,nu)>=idx(lam,sig) with idx(p,q)=p(p+1)/2+q; for those
# mu is the largest index, so each thread writes only its own eri[mu,...] rows.
eri = zeros(n_basis, n_basis, n_basis, n_basis)
parallel for mu to n_basis {
    for nu to n_basis {
        for lam to n_basis {
            for sig to n_basis {
                canon = false
                if mu >= nu {
                    if lam >= sig {
                        mn = mu * (mu + 1.0) / 2.0 + nu
                        ls = lam * (lam + 1.0) / 2.0 + sig
                        if mn >= ls { canon = true }
                    }
                }
                if canon {
                    eri[mu, nu, lam, sig] = cont_eri(mu, nu, lam, sig)
                }
            }
        }
    }
}
# Pass 2 (serial, cheap): scatter each canonical value to its images. For a
# canonical tuple the sort below is the identity, so the copy is a harmless no-op.
for mu to n_basis {
    for nu to n_basis {
        for lam to n_basis {
            for sig to n_basis {
                a = mu
                b = nu
                if nu > mu {
                    a = nu
                    b = mu
                }
                c = lam
                d = sig
                if sig > lam {
                    c = sig
                    d = lam
                }
                ab = a * (a + 1.0) / 2.0 + b
                cd = c * (c + 1.0) / 2.0 + d
                if cd > ab {
                    ta = a
                    tb = b
                    a = c
                    b = d
                    c = ta
                    d = tb
                }
                eri[mu, nu, lam, sig] = eri[a, b, c, d]
            }
        }
    }
}

# Symmetric orthogonalization X = S^{-1/2}.
s_vals = zeros(n_basis)
s_vecs = zeros(n_basis, n_basis)
eig_sym(S, s_vals, s_vecs)
D_inv_sqrt = zeros(n_basis, n_basis)
for i to n_basis {
    D_inv_sqrt[i, i] = 1.0 / sqrt(s_vals[i])
}
X = s_vecs @ D_inv_sqrt @ transpose(s_vecs)

# Nuclear repulsion.
E_nuc = 0.0
for i to n_atoms {
    for j to n_atoms {
        if j > i {
            dx = atom_x[i] - atom_x[j]
            dy = atom_y[i] - atom_y[j]
            dz = atom_z[i] - atom_z[j]
            E_nuc += atom_Z[i] * atom_Z[j] / sqrt(dx * dx + dy * dy + dz * dz)
        }
    }
}

# Solve the linear system A x = b (n unknowns) by Gaussian elimination with
# partial pivoting. A and b are overwritten in place; the solution lands in x.
# (Used to solve the small DIIS coefficient system each SCF cycle.)
def solve_linear(A, b, x, n) {
    for k to n {
        piv = k
        big = A[k, k]
        if big < 0.0 { big = -big }
        for i to n {
            if i > k {
                ai = A[i, k]
                if ai < 0.0 { ai = -ai }
                if ai > big {
                    big = ai
                    piv = i
                }
            }
        }
        if piv > k {
            for j to n {
                tmp = A[k, j]
                A[k, j] = A[piv, j]
                A[piv, j] = tmp
            }
            tb = b[k]
            b[k] = b[piv]
            b[piv] = tb
        }
        for i to n {
            if i > k {
                f = A[i, k] / A[k, k]
                for j to n {
                    A[i, j] = A[i, j] - f * A[k, j]
                }
                b[i] = b[i] - f * b[k]
            }
        }
    }
    for kk to n {
        i = n - 1.0 - kk
        s = b[i]
        for j to n {
            if j > i {
                s = s - A[i, j] * x[j]
            }
        }
        x[i] = s / A[i, i]
    }
}

# Robust SCF: run from two initial guesses and keep the lower (ground-state)
# energy. No single guess is universally safe -- the bare core guess sends N2 to
# a wrong excited solution, while the GWH guess gives the Be atom the wrong
# (1s^2 2p^2) occupation. Both are valid SCF fixed points, so by the variational
# principle the true ground state is simply the lower of the two converged energies.
#   guess 0: core (F0 = H_core)
#   guess 1: GWH  (F0_ii = H_ii, F0_ij = 0.875*(H_ii+H_jj)*S_ij)
E_best = 0.0
have_best = false
for guess to 2 {
    F_guess = zeros(n_basis, n_basis)
    for mu to n_basis {
        for nu to n_basis {
            if guess == 0.0 {
                F_guess[mu, nu] = H_core[mu, nu]
            }
            else {
                if mu == nu {
                    F_guess[mu, nu] = H_core[mu, mu]
                }
                else {
                    F_guess[mu, nu] = 0.875 * (H_core[mu, mu] + H_core[nu, nu]) * S[mu, nu]
                }
            }
        }
    }
Fp = transpose(X) @ F_guess @ X
eps_vals = zeros(n_basis)
Cp = zeros(n_basis, n_basis)
eig_sym(Fp, eps_vals, Cp)
C = X @ Cp
P = zeros(n_basis, n_basis)
for mu to n_basis {
    for nu to n_basis {
        psum = 0.0
        for a to n_occ {
            psum += C[mu, a] * C[nu, a]
        }
        P[mu, nu] = 2.0 * psum
    }
}

# SCF accelerated by DIIS (Pulay). The DIIS error e = X^T(FPS - SPF)X is the
# occupied-virtual gradient (zero at convergence); DIIS extrapolates the Fock
# matrix from a history of (F_i, e_i) pairs by minimizing the error in their
# span, converging in far fewer cycles than damped iteration.
max_diis = 8
diis_F = zeros(max_diis, n_basis, n_basis)
diis_E = zeros(max_diis, n_basis, n_basis)
n_diis = 0.0
E_total = 0.0
for iter to 100 {
    G = zeros(n_basis, n_basis)
    for mu to n_basis {
        for nu to n_basis {
            g = 0.0
            for lam to n_basis {
                for sig to n_basis {
                    g += P[lam, sig] * (eri[mu, nu, lam, sig] - 0.5 * eri[mu, lam, nu, sig])
                }
            }
            G[mu, nu] = g
        }
    }
    F = H_core + G
    # Electronic energy from the current density and its Fock.
    E_elec = 0.0
    for mu to n_basis {
        for nu to n_basis {
            E_elec += 0.5 * P[mu, nu] * (H_core[mu, nu] + F[mu, nu])
        }
    }
    E_total = E_elec + E_nuc
    # DIIS error matrix in the orthonormal basis, and its max element.
    err_ao = (F @ P @ S) - (S @ P @ F)
    err = transpose(X) @ err_ao @ X
    maxerr = 0.0
    for mu to n_basis {
        for nu to n_basis {
            ee = err[mu, nu]
            if ee < 0.0 { ee = -ee }
            if ee > maxerr { maxerr = ee }
        }
    }
    if maxerr < 1.0e-9 { break }
    # Push (F, err) onto the history, shifting out the oldest entry when full.
    if n_diis == max_diis {
        for k to (max_diis - 1.0) {
            for mu to n_basis {
                for nu to n_basis {
                    diis_F[k, mu, nu] = diis_F[k + 1.0, mu, nu]
                    diis_E[k, mu, nu] = diis_E[k + 1.0, mu, nu]
                }
            }
        }
    }
    else {
        n_diis += 1.0
    }
    slot = n_diis - 1.0
    for mu to n_basis {
        for nu to n_basis {
            diis_F[slot, mu, nu] = F[mu, nu]
            diis_E[slot, mu, nu] = err[mu, nu]
        }
    }
    # Build and solve the DIIS system B c = rhs (dimension n_diis + 1), then
    # replace F by the extrapolation sum_i c_i F_i.
    if n_diis > 1.0 {
        nb = n_diis + 1.0
        B = zeros(nb, nb)
        rhs = zeros(nb)
        for ai to n_diis {
            for bi to n_diis {
                bsum = 0.0
                for mu to n_basis {
                    for nu to n_basis {
                        bsum += diis_E[ai, mu, nu] * diis_E[bi, mu, nu]
                    }
                }
                B[ai, bi] = bsum
            }
        }
        for ai to n_diis {
            B[ai, n_diis] = -1.0
            B[n_diis, ai] = -1.0
        }
        rhs[n_diis] = -1.0
        cvec = zeros(nb)
        solve_linear(B, rhs, cvec, nb)
        # A singular B has no coefficients (solve_linear divides 0 by 0): the
        # error vectors of an atom like Be, whose empty 2p shell stays empty by
        # symmetry, are all but zero. Then the plain F is kept for this step;
        # a NaN in F would stop eig_sym.
        usable = 1.0
        for ai to nb {
            if cvec[ai] != cvec[ai] { usable = 0.0 }
        }
        if usable > 0.0 {
            F = zeros(n_basis, n_basis)
            for ai to n_diis {
                for mu to n_basis {
                    for nu to n_basis {
                        F[mu, nu] = F[mu, nu] + cvec[ai] * diis_F[ai, mu, nu]
                    }
                }
            }
        }
    }
    # New density from the (extrapolated) Fock.
    Fp = transpose(X) @ F @ X
    eps_vals = zeros(n_basis)
    Cp = zeros(n_basis, n_basis)
    eig_sym(Fp, eps_vals, Cp)
    C = X @ Cp
    P = zeros(n_basis, n_basis)
    for mu to n_basis {
        for nu to n_basis {
            psum = 0.0
            for a to n_occ {
                psum += C[mu, a] * C[nu, a]
            }
            P[mu, nu] = 2.0 * psum
        }
    }
    }
    # Keep the lower-energy (ground-state) solution across the two guesses, and
    # save its MOs/orbital energies (a fresh allocation per guess) for MP2.
    keep = false
    if not have_best {
        have_best = true
        keep = true
    }
    else {
        if E_total < E_best {
            keep = true
        }
    }
    if keep {
        E_best = E_total
        C_best = C
        eps_best = eps_vals
    }
}
E_total = E_best

# =====================================================================
# MP2 correlation energy on the converged RHF reference (closed-shell).
# Transform the AO integrals (chemist notation (mu nu|lam sig)) to the MO basis
# in four quarter-transforms, then sum the MP2 pair energies. O(N^5) per step.
# =====================================================================
mo1 = zeros(n_basis, n_basis, n_basis, n_basis)
for p to n_basis {
    for nu to n_basis {
        for lam to n_basis {
            for sig to n_basis {
                s = 0.0
                for mu to n_basis {
                    s += C_best[mu, p] * eri[mu, nu, lam, sig]
                }
                mo1[p, nu, lam, sig] = s
            }
        }
    }
}
mo2 = zeros(n_basis, n_basis, n_basis, n_basis)
for p to n_basis {
    for q to n_basis {
        for lam to n_basis {
            for sig to n_basis {
                s = 0.0
                for nu to n_basis {
                    s += C_best[nu, q] * mo1[p, nu, lam, sig]
                }
                mo2[p, q, lam, sig] = s
            }
        }
    }
}
mo3 = zeros(n_basis, n_basis, n_basis, n_basis)
for p to n_basis {
    for q to n_basis {
        for r to n_basis {
            for sig to n_basis {
                s = 0.0
                for lam to n_basis {
                    s += C_best[lam, r] * mo2[p, q, lam, sig]
                }
                mo3[p, q, r, sig] = s
            }
        }
    }
}
gmo = zeros(n_basis, n_basis, n_basis, n_basis)
for p to n_basis {
    for q to n_basis {
        for r to n_basis {
            for t to n_basis {
                s = 0.0
                for sig to n_basis {
                    s += C_best[sig, t] * mo3[p, q, r, sig]
                }
                gmo[p, q, r, t] = s
            }
        }
    }
}
# E_corr = sum_{ij occ, ab virt} (ia|jb)[2(ia|jb) - (ib|ja)] / (e_i + e_j - e_a - e_b)
emp2 = 0.0
for oi to n_occ {
    for oj to n_occ {
        for va to n_basis {
            if va >= n_occ {
                for vb to n_basis {
                    if vb >= n_occ {
                        iajb = gmo[oi, va, oj, vb]
                        ibja = gmo[oi, vb, oj, va]
                        denom = eps_best[oi] + eps_best[oj] - eps_best[va] - eps_best[vb]
                        emp2 += iajb * (2.0 * iajb - ibja) / denom
                    }
                }
            }
        }
    }
}
e_mp2 = E_total + emp2

show E_total
show emp2
show e_mp2

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