Notebooks · lib/radial

The radial Schrodinger equation, the way dftatom solves it

An atom's orbitals are one-dimensional. Separate the angular part out and what is left is

u'' = [ l(l+1)/r^2 + 2(V(r) - E) ] u,    u = r R(r),

on 0 < r < infinity, with u(0) = 0 and u -> 0 at infinity. Two boundary conditions and one unknown E too many, which is what makes it an eigenvalue problem: only particular energies admit a solution that vanishes at both ends.

The mesh. The 1s of uranium lives inside a hundredth of a Bohr while the outer shells run to tens, so the mesh is exponential -- r = alpha (e^(beta t) - 1) + r_min on a uniform t. The one used here is dftatom's own: r_min = 1e-8, r_max = 50, a ratio of 1e6 between the last interval and the first, 3000 intervals.

The integrator. Changing variable from r to t puts a first derivative into the equation, and Numerov's method cannot have one. Substituting w = u / sqrt(dr/dt) takes it back out, at the cost of one extra term which for an exponential mesh is the constant beta^2/4:

w'' = [ (dr/dt)^2 ( l(l+1)/r^2 + 2(V - E) ) + beta^2/4 ] w.

That is Numerov's form on a uniform grid: fourth order, and it reads only mesh points, so no interpolation is needed between them.

The shooting. Integrate out from r_min, where u ~ r^(l+1), and in from r_max, where u ~ e^(-sqrt(-2E) r). Each direction is integrated the way it grows, which is the stable way to do it. They meet at the classical turning point; where the two logarithmic derivatives agree, E is an eigenvalue. Node counting brackets it and the derivative mismatch refines it.

The test is dftatom's own: hydrogen-like uranium, Z = 92, where every eigenvalue is known exactly to be -Z^2/2n^2. It is chosen because it is hard. The 1s state sits at -4232 Hartree and is squeezed into the first thousandth of the mesh.

Used with import radial.

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[1]lines 44–357

import linalg

#| The iteration limits of the eigenvalue search. Each loop stops early, on its
#| own test; these only keep a search that has lost its bracket from running
#| forever. Bisections of a band boundary, steps widening the bracket
#| downwards, and bisections of the bracketed eigenvalue.
RADIAL_BAND_BISECTIONS = 200
RADIAL_BRACKET_STEPS = 300
RADIAL_BISECTIONS = 300

#| dftatom's exponential mesh: `N + 1` points from `r_min` to `r_max`, with the
#| last interval `a` times the first.
def mesh_exp(r_min, r_max, a, N) {
    beta = log(a) / (N - 1.0)
    alpha = (r_max - r_min) / (exp(beta * N) - 1.0)
    out = zeros(N + 1)
    for i to N + 1 {
        out[i] = alpha * (exp(beta * i) - 1.0) + r_min
    }
    return out
}

#| `dr/dt` on that mesh, with `t` the uniform index.
def mesh_exp_deriv(r_min, r_max, a, N) {
    beta = log(a) / (N - 1.0)
    alpha = (r_max - r_min) / (exp(beta * N) - 1.0)
    out = zeros(N + 1)
    for i to N + 1 {
        out[i] = alpha * beta * exp(beta * i)
    }
    return out
}

#| The `beta` of a mesh built by `mesh_exp`, which the Numerov substitution
#| needs.
def mesh_exp_beta(a, N) {
    return log(a) / (N - 1.0)
}

#| The coefficient `f` in `w'' = f w`, at every mesh point.
def radial_f(R: Arr, Rp: Arr, V: Arr, l, E, beta, np) {
    f = zeros(np)
    for i to np {
        centrifugal = l * (l + 1.0) / (R[i] * R[i])
        f[i] = Rp[i] * Rp[i] * (centrifugal + 2.0 * (V[i] - E)) + 0.25 * beta * beta
    }
    return f
}

#| One Numerov step, from `w[i-1]` and `w[i]` to `w[i+1]`, with unit spacing.
def numerov_next(wm, w0, fm, f0, fp) {
    return ((2.0 + 5.0 * f0 / 6.0) * w0 - (1.0 - fm / 12.0) * wm) / (1.0 - fp / 12.0)
}

#| Integrate outward from the origin to `imax`, filling `w`. Returns how many
#| times the solution crossed zero, which is the quantum number the state will
#| turn out to have.
#|
#| The solution grows by many orders of magnitude on the way out, so it is
#| rescaled whenever it gets large. Only ratios are ever asked of it.
def radial_outward(R: Arr, Rp: Arr, f: Arr, l, imax, w: Arr) {
    w[0] = pow(R[0], l + 1.0) / sqrt(Rp[0])
    w[1] = pow(R[1], l + 1.0) / sqrt(Rp[1])
    nodes = 0
    for k to imax - 1 {
        i = k + 1
        w[i + 1] = numerov_next(w[i - 1], w[i], f[i - 1], f[i], f[i + 1])
        if w[i + 1] * w[i] < 0.0 {
            nodes = nodes + 1
        }
        if abs(w[i + 1]) > 1.0e100 {
            for j to i + 2 {
                w[j] = w[j] * 1.0e-100
            }
        }
    }
    return 1.0 * nodes
}

#| Integrate inward from the outer edge to `imin`. Inward is the direction this
#| solution grows in, which is why it is done this way round.
def radial_inward(R: Arr, Rp: Arr, f: Arr, E, imin, np, w: Arr) {
    lam = sqrt(-2.0 * E)
    last = np - 1
    w[last] = 1.0 / sqrt(Rp[last])
    w[last - 1] = exp(-lam * (R[last - 1] - R[last])) / sqrt(Rp[last - 1])
    nodes = 0
    for k to last - imin - 1 {
        i = last - 1 - k
        w[i - 1] = numerov_next(w[i + 1], w[i], f[i + 1], f[i], f[i - 1])
        if w[i - 1] * w[i] < 0.0 {
            nodes = nodes + 1
        }
        if abs(w[i - 1]) > 1.0e100 {
            for j to last - i + 2 {
                w[last - j] = w[last - j] * 1.0e-100
            }
        }
    }
    return 1.0 * nodes
}

#| The outermost point where the classical kinetic energy is still positive.
#| Matching there keeps both integrations in the region where they are growing.
def radial_turning(R: Arr, V: Arr, l, E, np) {
    best = 2
    for i to np {
        eff = V[i] + l * (l + 1.0) / (2.0 * R[i] * R[i])
        if eff < E {
            best = i
        }
    }
    if best < 2 {
        best = 2
    }
    if best > np - 3 {
        best = np - 3
    }
    return best
}

#| How badly the two solutions disagree at the matching point, in logarithmic
#| derivative. A zero of this, at the right node count, is an eigenvalue.
def radial_mismatch(R: Arr, Rp: Arr, V: Arr, l, E, beta, np, wout: Arr, win: Arr) {
    f = radial_f(R, Rp, V, l, E, beta, np)
    im = radial_turning(R, V, l, E, np)
    junk = radial_outward(R, Rp, f, l, im + 1, wout)
    junk = radial_inward(R, Rp, f, E, im - 1, np, win)
    dout = (wout[im + 1] - wout[im - 1]) / (2.0 * wout[im])
    din = (win[im + 1] - win[im - 1]) / (2.0 * win[im])
    return dout - din
}

#| How many times the outward solution crosses zero inside the classically
#| allowed region. This is `n - l - 1` for the state the energy belongs to, and
#| it rises by one at every eigenvalue, so it says which state a trial energy is
#| nearest without locating it.
#|
#| It has to stop at the turning point. Past it the equation is
#| `w'' = f w` with `f` large and positive -- 450 for hydrogen-like uranium at
#| the outer edge -- and Numerov needs `|f| < 12` to be stable at all. Counting
#| across the whole mesh returns hundreds of nodes, all of them the integrator
#| oscillating on its own.
def radial_nodes(R: Arr, Rp: Arr, V: Arr, l, E, beta, np, w: Arr) {
    f = radial_f(R, Rp, V, l, E, beta, np)
    im = radial_turning(R, V, l, E, np)
    return radial_outward(R, Rp, f, l, im, w)
}

#| The lowest energy in `[Elo, Ehi]` whose solution has more than `k` nodes
#| inside the classically allowed region: the top of the band that belongs to
#| the state with `k` nodes.
def radial_band_top(R: Arr, Rp: Arr, V: Arr, l, k, Elo, Ehi, beta, np, w: Arr) {
    lo = Elo
    hi = Ehi
    for it to RADIAL_BAND_BISECTIONS {
        mid = 0.5 * (lo + hi)
        if radial_nodes(R, Rp, V, l, mid, beta, np, w) > k {
            hi = mid
        }
        else {
            lo = mid
        }
        # a band boundary only has to be identified, not resolved: the mismatch
        # does the locating, and every digit spent here is an integration wasted
        if hi - lo < 1.0e-6 * (abs(hi) + 1.0) {
            break
        }
    }
    return 0.5 * (lo + hi)
}

#| The lowest the effective potential goes. No state can lie below it, and no
#| energy below it has a classically allowed region at all -- which for `l = 2`
#| in hydrogen-like uranium is everything below -352, because the centrifugal
#| barrier `l(l+1)/2r^2` lifts the well that far. Searching from an arbitrary
#| floor like -6000 puts every d and f state outside the region the turning
#| point is defined in, and they come back wrong.
def radial_floor(R: Arr, V: Arr, l, np) {
    lowest = 1.0e30
    for i to np {
        lowest = min(lowest, V[i] + l * (l + 1.0) / (2.0 * R[i] * R[i]))
    }
    return lowest
}

#| Solve for the eigenvalue of state `n, l` in the potential `V`.
#|
#| Two signals, doing two different jobs, and it matters which is asked what.
#|
#| The node count says which state a trial energy belongs to. It carves the
#| spectrum into one band per state. Its boundaries are NOT the eigenvalues --
#| for hydrogen-like uranium the count goes from zero to one near -3200 while
#| the 1s sits at -4232 -- so it can say "this band holds the 1s" and nothing
#| more precise than that.
#|
#| The logarithmic-derivative mismatch locates the state. Inside one band it is
#| smooth with exactly one zero, and that zero is the eigenvalue. So: the node
#| count finds the top of the band, the mismatch is stepped downward from there
#| until it changes sign, and bisection finishes.
def radial_solve(R: Arr, Rp: Arr, V: Arr, n, l, Ehi, beta, np) {
    want = n - l - 1.0
    wout = zeros(np)
    win = zeros(np)
    floor = radial_floor(R, V, l, np)
    guard = floor + 1.0e-9 * abs(floor)

    # the top of this state's band, and the bottom of it if there is one
    top = radial_band_top(R, Rp, V, l, want, guard, Ehi, beta, np, wout)
    limit = guard
    if want > 0.0 {
        limit = radial_band_top(R, Rp, V, l, want - 1.0, guard, Ehi, beta, np, wout)
    }

    b = top - 1.0e-4 * abs(top)
    fb = radial_mismatch(R, Rp, V, l, b, beta, np, wout, win)
    a = b
    fa = fb
    step = max(1.0e-9, 1.0e-3 * abs(b))
    for it to RADIAL_BRACKET_STEPS {
        a = a - step
        if a < limit {
            a = limit + 1.0e-6 * abs(limit)
        }
        fa = radial_mismatch(R, Rp, V, l, a, beta, np, wout, win)
        if fa * fb <= 0.0 {
            break
        }
        b = a
        fb = fa
        step = step * 1.5
        if a <= limit + 2.0e-6 * abs(limit) {
            break
        }
    }
    if fa * fb > 0.0 {
        return a
    }
    for it to RADIAL_BISECTIONS {
        mid = 0.5 * (a + b)
        fm = radial_mismatch(R, Rp, V, l, mid, beta, np, wout, win)
        if fa * fm <= 0.0 {
            b = mid
            fb = fm
        }
        else {
            a = mid
            fa = fm
        }
        if abs(b - a) < 1.0e-14 * (abs(a) + 1.0) {
            break
        }
    }
    return 0.5 * (a + b)
}

test "the mesh is dftatom's, endpoints and ratio included" {
    N = 3000
    R = mesh_exp(1.0e-8, 50.0, 1.0e6, N)
    assert_near(R[0], 1.0e-8, 1.0e-18)
    assert_near(R[N], 50.0, 1.0e-12)
    # the last interval is a million times the first, which is what `a` means
    first = R[1] - R[0]
    last = R[N] - R[N - 1]
    assert_near(last / first, 1.0e6, 1.0)
    # and it ascends
    for i to N {
        assert_lt(R[i], R[i + 1])
    }
}

test "the mesh derivative is the derivative of the mesh" {
    N = 500
    R = mesh_exp(1.0e-6, 20.0, 1.0e4, N)
    Rp = mesh_exp_deriv(1.0e-6, 20.0, 1.0e4, N)
    beta = mesh_exp_beta(1.0e4, N)
    # A central difference of an exponential is not exact; it is out by
    # beta^2/6 relatively, and that is what the gap should be. Asserting a
    # tolerance instead would be asserting how bad the check is.
    npts = 40
    for k to npts {
        i = 5 + 12 * k
        fd = 0.5 * (R[i + 1] - R[i - 1])
        rel = (fd - Rp[i]) / Rp[i]
        assert_near(rel, beta * beta / 6.0, 1.0e-9)
    }
}

test "for hydrogen-like uranium every level is minus Z squared over two n squared, to a relative 1e-8" {
    Z = 92.0
    N = 3000
    R = mesh_exp(1.0e-8, 50.0, 1.0e6, N)
    Rp = mesh_exp_deriv(1.0e-8, 50.0, 1.0e6, N)
    beta = mesh_exp_beta(1.0e6, N)
    np = N + 1
    V = zeros(np)
    for i to np {
        V[i] = -Z / R[i]
    }
    worst = 0.0
    nshells = 7
    for n to nshells {
        nn = n + 1.0
        exact = -Z * Z / (2.0 * nn * nn)
        for l to nn {
            e = radial_solve(R, Rp, V, nn, 1.0 * l, -1.0e-6, beta, np)
            worst = max(worst, abs(e - exact) / abs(exact))
        }
    }
    assert_lt(worst, 1.0e-8)
}

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

import linalg

#| The iteration limits of the eigenvalue search. Each loop stops early, on its
#| own test; these only keep a search that has lost its bracket from running
#| forever. Bisections of a band boundary, steps widening the bracket
#| downwards, and bisections of the bracketed eigenvalue.
RADIAL_BAND_BISECTIONS = 200
RADIAL_BRACKET_STEPS = 300
RADIAL_BISECTIONS = 300

#| dftatom's exponential mesh: `N + 1` points from `r_min` to `r_max`, with the
#| last interval `a` times the first.
def mesh_exp(r_min, r_max, a, N) {
    beta = log(a) / (N - 1.0)
    alpha = (r_max - r_min) / (exp(beta * N) - 1.0)
    out = zeros(N + 1)
    for i to N + 1 {
        out[i] = alpha * (exp(beta * i) - 1.0) + r_min
    }
    return out
}

#| `dr/dt` on that mesh, with `t` the uniform index.
def mesh_exp_deriv(r_min, r_max, a, N) {
    beta = log(a) / (N - 1.0)
    alpha = (r_max - r_min) / (exp(beta * N) - 1.0)
    out = zeros(N + 1)
    for i to N + 1 {
        out[i] = alpha * beta * exp(beta * i)
    }
    return out
}

#| The `beta` of a mesh built by `mesh_exp`, which the Numerov substitution
#| needs.
def mesh_exp_beta(a, N) {
    return log(a) / (N - 1.0)
}

#| The coefficient `f` in `w'' = f w`, at every mesh point.
def radial_f(R: Arr, Rp: Arr, V: Arr, l, E, beta, np) {
    f = zeros(np)
    for i to np {
        centrifugal = l * (l + 1.0) / (R[i] * R[i])
        f[i] = Rp[i] * Rp[i] * (centrifugal + 2.0 * (V[i] - E)) + 0.25 * beta * beta
    }
    return f
}

#| One Numerov step, from `w[i-1]` and `w[i]` to `w[i+1]`, with unit spacing.
def numerov_next(wm, w0, fm, f0, fp) {
    return ((2.0 + 5.0 * f0 / 6.0) * w0 - (1.0 - fm / 12.0) * wm) / (1.0 - fp / 12.0)
}

#| Integrate outward from the origin to `imax`, filling `w`. Returns how many
#| times the solution crossed zero, which is the quantum number the state will
#| turn out to have.
#|
#| The solution grows by many orders of magnitude on the way out, so it is
#| rescaled whenever it gets large. Only ratios are ever asked of it.
def radial_outward(R: Arr, Rp: Arr, f: Arr, l, imax, w: Arr) {
    w[0] = pow(R[0], l + 1.0) / sqrt(Rp[0])
    w[1] = pow(R[1], l + 1.0) / sqrt(Rp[1])
    nodes = 0
    for k to imax - 1 {
        i = k + 1
        w[i + 1] = numerov_next(w[i - 1], w[i], f[i - 1], f[i], f[i + 1])
        if w[i + 1] * w[i] < 0.0 {
            nodes = nodes + 1
        }
        if abs(w[i + 1]) > 1.0e100 {
            for j to i + 2 {
                w[j] = w[j] * 1.0e-100
            }
        }
    }
    return 1.0 * nodes
}

#| Integrate inward from the outer edge to `imin`. Inward is the direction this
#| solution grows in, which is why it is done this way round.
def radial_inward(R: Arr, Rp: Arr, f: Arr, E, imin, np, w: Arr) {
    lam = sqrt(-2.0 * E)
    last = np - 1
    w[last] = 1.0 / sqrt(Rp[last])
    w[last - 1] = exp(-lam * (R[last - 1] - R[last])) / sqrt(Rp[last - 1])
    nodes = 0
    for k to last - imin - 1 {
        i = last - 1 - k
        w[i - 1] = numerov_next(w[i + 1], w[i], f[i + 1], f[i], f[i - 1])
        if w[i - 1] * w[i] < 0.0 {
            nodes = nodes + 1
        }
        if abs(w[i - 1]) > 1.0e100 {
            for j to last - i + 2 {
                w[last - j] = w[last - j] * 1.0e-100
            }
        }
    }
    return 1.0 * nodes
}

#| The outermost point where the classical kinetic energy is still positive.
#| Matching there keeps both integrations in the region where they are growing.
def radial_turning(R: Arr, V: Arr, l, E, np) {
    best = 2
    for i to np {
        eff = V[i] + l * (l + 1.0) / (2.0 * R[i] * R[i])
        if eff < E {
            best = i
        }
    }
    if best < 2 {
        best = 2
    }
    if best > np - 3 {
        best = np - 3
    }
    return best
}

#| How badly the two solutions disagree at the matching point, in logarithmic
#| derivative. A zero of this, at the right node count, is an eigenvalue.
def radial_mismatch(R: Arr, Rp: Arr, V: Arr, l, E, beta, np, wout: Arr, win: Arr) {
    f = radial_f(R, Rp, V, l, E, beta, np)
    im = radial_turning(R, V, l, E, np)
    junk = radial_outward(R, Rp, f, l, im + 1, wout)
    junk = radial_inward(R, Rp, f, E, im - 1, np, win)
    dout = (wout[im + 1] - wout[im - 1]) / (2.0 * wout[im])
    din = (win[im + 1] - win[im - 1]) / (2.0 * win[im])
    return dout - din
}

#| How many times the outward solution crosses zero inside the classically
#| allowed region. This is `n - l - 1` for the state the energy belongs to, and
#| it rises by one at every eigenvalue, so it says which state a trial energy is
#| nearest without locating it.
#|
#| It has to stop at the turning point. Past it the equation is
#| `w'' = f w` with `f` large and positive -- 450 for hydrogen-like uranium at
#| the outer edge -- and Numerov needs `|f| < 12` to be stable at all. Counting
#| across the whole mesh returns hundreds of nodes, all of them the integrator
#| oscillating on its own.
def radial_nodes(R: Arr, Rp: Arr, V: Arr, l, E, beta, np, w: Arr) {
    f = radial_f(R, Rp, V, l, E, beta, np)
    im = radial_turning(R, V, l, E, np)
    return radial_outward(R, Rp, f, l, im, w)
}

#| The lowest energy in `[Elo, Ehi]` whose solution has more than `k` nodes
#| inside the classically allowed region: the top of the band that belongs to
#| the state with `k` nodes.
def radial_band_top(R: Arr, Rp: Arr, V: Arr, l, k, Elo, Ehi, beta, np, w: Arr) {
    lo = Elo
    hi = Ehi
    for it to RADIAL_BAND_BISECTIONS {
        mid = 0.5 * (lo + hi)
        if radial_nodes(R, Rp, V, l, mid, beta, np, w) > k {
            hi = mid
        }
        else {
            lo = mid
        }
        # a band boundary only has to be identified, not resolved: the mismatch
        # does the locating, and every digit spent here is an integration wasted
        if hi - lo < 1.0e-6 * (abs(hi) + 1.0) {
            break
        }
    }
    return 0.5 * (lo + hi)
}

#| The lowest the effective potential goes. No state can lie below it, and no
#| energy below it has a classically allowed region at all -- which for `l = 2`
#| in hydrogen-like uranium is everything below -352, because the centrifugal
#| barrier `l(l+1)/2r^2` lifts the well that far. Searching from an arbitrary
#| floor like -6000 puts every d and f state outside the region the turning
#| point is defined in, and they come back wrong.
def radial_floor(R: Arr, V: Arr, l, np) {
    lowest = 1.0e30
    for i to np {
        lowest = min(lowest, V[i] + l * (l + 1.0) / (2.0 * R[i] * R[i]))
    }
    return lowest
}

#| Solve for the eigenvalue of state `n, l` in the potential `V`.
#|
#| Two signals, doing two different jobs, and it matters which is asked what.
#|
#| The node count says which state a trial energy belongs to. It carves the
#| spectrum into one band per state. Its boundaries are NOT the eigenvalues --
#| for hydrogen-like uranium the count goes from zero to one near -3200 while
#| the 1s sits at -4232 -- so it can say "this band holds the 1s" and nothing
#| more precise than that.
#|
#| The logarithmic-derivative mismatch locates the state. Inside one band it is
#| smooth with exactly one zero, and that zero is the eigenvalue. So: the node
#| count finds the top of the band, the mismatch is stepped downward from there
#| until it changes sign, and bisection finishes.
def radial_solve(R: Arr, Rp: Arr, V: Arr, n, l, Ehi, beta, np) {
    want = n - l - 1.0
    wout = zeros(np)
    win = zeros(np)
    floor = radial_floor(R, V, l, np)
    guard = floor + 1.0e-9 * abs(floor)

# the top of this state's band, and the bottom of it if there is one
    top = radial_band_top(R, Rp, V, l, want, guard, Ehi, beta, np, wout)
    limit = guard
    if want > 0.0 {
        limit = radial_band_top(R, Rp, V, l, want - 1.0, guard, Ehi, beta, np, wout)
    }

b = top - 1.0e-4 * abs(top)
    fb = radial_mismatch(R, Rp, V, l, b, beta, np, wout, win)
    a = b
    fa = fb
    step = max(1.0e-9, 1.0e-3 * abs(b))
    for it to RADIAL_BRACKET_STEPS {
        a = a - step
        if a < limit {
            a = limit + 1.0e-6 * abs(limit)
        }
        fa = radial_mismatch(R, Rp, V, l, a, beta, np, wout, win)
        if fa * fb <= 0.0 {
            break
        }
        b = a
        fb = fa
        step = step * 1.5
        if a <= limit + 2.0e-6 * abs(limit) {
            break
        }
    }
    if fa * fb > 0.0 {
        return a
    }
    for it to RADIAL_BISECTIONS {
        mid = 0.5 * (a + b)
        fm = radial_mismatch(R, Rp, V, l, mid, beta, np, wout, win)
        if fa * fm <= 0.0 {
            b = mid
            fb = fm
        }
        else {
            a = mid
            fa = fm
        }
        if abs(b - a) < 1.0e-14 * (abs(a) + 1.0) {
            break
        }
    }
    return 0.5 * (a + b)
}

test "the mesh is dftatom's, endpoints and ratio included" {
    N = 3000
    R = mesh_exp(1.0e-8, 50.0, 1.0e6, N)
    assert_near(R[0], 1.0e-8, 1.0e-18)
    assert_near(R[N], 50.0, 1.0e-12)
    # the last interval is a million times the first, which is what `a` means
    first = R[1] - R[0]
    last = R[N] - R[N - 1]
    assert_near(last / first, 1.0e6, 1.0)
    # and it ascends
    for i to N {
        assert_lt(R[i], R[i + 1])
    }
}

test "the mesh derivative is the derivative of the mesh" {
    N = 500
    R = mesh_exp(1.0e-6, 20.0, 1.0e4, N)
    Rp = mesh_exp_deriv(1.0e-6, 20.0, 1.0e4, N)
    beta = mesh_exp_beta(1.0e4, N)
    # A central difference of an exponential is not exact; it is out by
    # beta^2/6 relatively, and that is what the gap should be. Asserting a
    # tolerance instead would be asserting how bad the check is.
    npts = 40
    for k to npts {
        i = 5 + 12 * k
        fd = 0.5 * (R[i + 1] - R[i - 1])
        rel = (fd - Rp[i]) / Rp[i]
        assert_near(rel, beta * beta / 6.0, 1.0e-9)
    }
}

test "for hydrogen-like uranium every level is minus Z squared over two n squared, to a relative 1e-8" {
    Z = 92.0
    N = 3000
    R = mesh_exp(1.0e-8, 50.0, 1.0e6, N)
    Rp = mesh_exp_deriv(1.0e-8, 50.0, 1.0e6, N)
    beta = mesh_exp_beta(1.0e6, N)
    np = N + 1
    V = zeros(np)
    for i to np {
        V[i] = -Z / R[i]
    }
    worst = 0.0
    nshells = 7
    for n to nshells {
        nn = n + 1.0
        exact = -Z * Z / (2.0 * nn * nn)
        for l to nn {
            e = radial_solve(R, Rp, V, nn, 1.0 * l, -1.0e-6, beta, np)
            worst = max(worst, abs(e - exact) / abs(exact))
        }
    }
    assert_lt(worst, 1.0e-8)
}

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