Chemical Compound Design Using Nuclear 

Charge Distributions

by B. Christopher Rinderspacher

ARL-TR-5935 March 2012

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005

ARL-TR-5935 March 2012

Chemical Compound Design Using Nuclear 

Charge Distributions

B. Christopher Rinderspacher

Weapons and Materials Research Directorate, ARL

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Chemical Compound Design Using Nuclear Charge Distributions

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14. ABSTRACT

Finding optimal solutions to design problems in chemistry is hampered by the combinatorially large search space. We 

develop a general theoretical framework for finding chemical compounds with prescribed properties using nuclear charge 

distributions. The key is the reformulation of the design problem into an optimization problem on probability distributions in 

chemical space. In order to achieve tractability, a constrained search formalism on the nuclear charge distributions, which are 

non-negative, is used to reduce the dimensionality of the problem. Furthermore, we introduce approximations to the exact 

functional, as derived, for common design properties and constraints.

15. SUBJECT TERMS

Chemical design, nuclear charge distribution

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Contents

1. Introduction 1

2. Distributions as Optimization Variables 1

3. Approximating O ̃ 6

4. Enforcing Physicality 7

5. Conclusions 9

References 10

Distribution 13

iii
iv

INTENTIONALLY LEFT BLANK
1. Introduction

Molecular inverse design beyond the purview of drug design has of late enjoyed increasing

popularity. Examples can be found in protein design (1–4) or high-hyperpolarizability

materials (5–7). The design problem is complicated by the vastness of possible chemicals,

termed chemical space. This space can be viewed as combinatorially complex, e.g., there

are 208 ≈ 2.6 · 1010 octa-peptides of the naturally occurring amino acids alone. As a

consequence, a variety of methods have been developed for discrete optimization in

chemical subspaces (8–14). All these methods restrict the search to a predetermined

subspace, often focused around molecular scaffolds. While this is convenient for reduced

computational cost, it represents a bias, which precludes the discovery of unsuspected and

truly novel solutions to the design problem, but also requires a priori knowledge to be

successful. Instead of restricting the space explicitly, a “functional” description of the

search space could remove the bias. Such a functional description must encode the target

properties and constraints of the problem in a fashion that is amenable to computation.

The continuous optimization of chemicals used in the

linear-combination-of-atomic-potentials (LCAP) method (15, 16) and the

variation-of-particles density-functional-theoretical (VP-DFT) method introduces an

important concept to overcoming the bias efficiently (17–19). LCAP and VP-DFT

interpolate continuously between the Hamiltonians of various chemical species.

Furthermore, recently an investigation into the reasons why chemical optimization is,

relatively speaking, “easy” used probability distributions and expectations on the control

variables to arrive at its conclusion (20, 21). It is this general idea that is extended and

developed in the following, employing distributions of nuclear charges and associated

potentials. Since the analytical form of such functionals is not generally known, we derive

approximate functionals for the optimization of an electronic property for a molecule in the

ground state (which is applied as a constraint), outlining a procedure for the development

of other functionals.

2. Distributions as Optimization Variables

Our goal is to develop an optimization algorithm of chemical species based on non-negative

distributions only. In order to be useful, it is required that the result can be identified

(preferably uniquely) with a specific chemical. As a consequence, it is necessary to describe

the properties of interest, the targets, and the constraints, as functionals in terms of these

distributions. This is accomplished via probability distributions on chemical space.

Furthermore, for properties with proper optima, the existence of and convergence on global

1
optima are proven for the exact functionals. We begin by defining rigorously the terms

chemical space, conformational space, and configurational space as used later.

Definition 1. The compositional space C, i.e., the space of chemical compounds, is

identified with the set of descendingly ordered finite sequences in N. Each non-zero entry of

the sequence denotes the charge of an atomic nucleus.

For example, lithium hydride (LiH) corresponds to (3, 1) and beryllium hydride (BeH2) to

(4, 1, 1). Due to the common restriction of charge, the charge is considered a separate

property derivable from the composition and problem setting rather than an inherent

component of the chemical definition, although it is possible to augment this picture

without altering the general applicability of the following results. This choice of identifying

the composition of chemicals reflects common usage in chemistry as well as providing a

unique mathematical object for each such composition.

Definition 2. The configurational space Γ is the set of finite sequences ((qi, Ri))i∈I in

N × R

3 of strictly descending lexicographical order (i < j if qi > qj or qi = qj, Rν

i = Rν

j

for

0 < ν < μ ≤ 3 and R

μ

i < Rμj

). The coordinate system is chosen to conform to the rules:

1. The coordinate system is right-handed and comprised of principal axes through the

center of mass.

2. If the coordinate system is ambiguous, then the axes are chosen to be a right-handed

system such that the resultant sequence is minimal with respect to lexicographical

order between sequences.

The second entry in each couplet is the position of the respective atom in real space.

Ambiguous coordinate systems may arise from high symmetry groups, which necessitates

the second rule in defining the coordinate system. Now, let γ : Γ → C map a configuration

to its respective composition, i.e., ((qi, Ri))i∈I 7→ (qi)i∈I.

Definition 3. The conformational space Ωc of a composition c is the pre-image

γ

−1

(c) ≡ Ωc ⊂ (R

3

)

`(c)

, where `(c) is the length of the sequence c.

For example, LiH corresponds to

ΩLiH = {(x, y, z) × (u, v, w) : x > u ∨ (x = u ∧ y > v)∨ (x = u ∧ y = v ∧ z > w)} ⊂ R

3 × R3

.

Each of the preceding definitions refines the general notions of chemistry rigorously, which

allows for well-defined treatment and analysis of chemical properties. We introduce the

following terminology to simplify expressions.

• Let C : N → C be an enumeration of compositions.

2
• The nuclear charge of C(i) is kC(i)k1 =

P`(C(i))

j=1 (C(i))j, e.g., for LiH, the charge is 4.

• Let Pi: γ

−1

(C(i)) ≡ ΩC(i) → R be a probability distribution on the conformational

space of composition C(i), i.e., Pi(x) ≥ 0∀x ∈ ΩC(i) and RPidμ(ΩC(i)) = 1 with

measure μ on ΩC(i).

• Let p

0

: C → R be a probability distribution on compositional space, which is

equivalent to p : N → R, p(i) = p

0

(C(i)), i.e., p(i) ≥ 0∀i ∈ N and P

i∈N

p(i) = 1.

For a probability distribution ρ : Γ → R, p and Pi can be defined such that the probability

finding composition C(i) at conformation Ri ∈ ΩC(i)is p(i)Pi(Ri), via

p(i) = R

γ−1(C(i)) ρ(x)dμ(Γ) and Pi

: γ

−1

(C(i)) → R, x 7→ ρ(x)/p(i).

With the help of these probability distributions, the following lemma now sets the stage for

optimizations in chemical space.

Lemma 1. For a bounded property O : Γ → R that attains its global optimum exactly once,

the optimal nuclear charge distribution is a distribution of discrete nuclei and charges.

Proof. Let Omin := O(Xmin). Then the expectation value for O is a linear functional of

probability distributions, and hOi

ρ =

R

Γ

ρ(x)O(x)dx has a minimal probability distribution

ρmin(x) = δΓ,Xmin(x). Here δΓ,X is the Dirac distribution on Γ, i.e., the probability

distribution on Γ such that δΓ,X(y) = 0∀y ∈ Γ\{X}. Hence, hOiρmin= O(Xmin) = Omin

and the minimum is attainable. Due to 0 ≤ | R

Γ

ρ(x)(O(x) − Omin)dx| ≤

R

Γ

|ρ(x)(O(x) − Omin)|dx =

R

Γ

ρ(x)(O(x) − Omin)dx = hOiρ − Omin = hOiρ − hOiρmin, there

is no other distribution that produces a lower expectation value. Since hOi

ρ

is linear and

the space of probability distributions is convex, the minimum is global. Applying the

argument to −O yields the same for maxima. Since Xmin is a single finite configuration,

the associated nuclear charge distribution is just

qXmin(R) =

`(X

Xmin)

j=1

qjδ

3

(R − Rj ), (1)

where δ

3

is the Dirac distribution on R

3

.

Analogously to equation 1, a nuclear charge distribution can be defined for any probability

distribution ρ on Γ. For simplicity, assume that Omin is non-degenerate and, without loss

of generality in the following, all compounds are considered to be neutral. We next clarify

nuclear charge distributions.

3
Definition 4. The nuclear charge distribution q(R) of a probability distribution ρ on

Γ with p and Pi is given by

qρ(R) := X

i

p(i)

Z

x=((qj ,Rj))`(C(i))

j=1 ∈γ−1(C(i))

Pi(x)

`(

X

C(i))

j=1

qjδ

3

(R − Rj )dμ(Γ).

The expectation charge of a distribution is q := R

R3 q(R)dR. A non-negative distribution q

is called configurationally representable if and only if there exists x ∈ Γ such that

q = qδΓ,x.

In order to perform optimizations over these distributions, it is helpful to understand the

structure of the distributions derived from probability distributions on Γ. The convex space

of potential nuclear charge distributions Q ⊂ L1(R

3

) is restricted to such distributions that

are non-negative on R

3

, whose coordinate systems coincide with the respective principal

axes and centers of mass, and for which q ≥ 1. Since the space of probability distributions

is convex and closed, so is the space of respective nuclear charge distributions QΓ ⊂ Q.

The only nuclear charge distribution with q = 1 is q(R) = δ(R), which corresponds to the

hydrogen atom. Hence, the set of non-negative distributions can be convexly decomposed

and the associated projection onto nuclear charge distributions is itself convex.

With this characterization it is possible to define unambiguously a functional on the space

of non-negative distributions for a given bounded property.

Definition 5.

O ̃[q] := 

minρ:qρ=q{hOi

ρ

}, ∃ρ : qρ = q

∞, otherwise

This functional and its properties are our main objective and the remainder of this section

is dedicated to characterizing this functional. The following lemma guarantees that the

minima of O ̃ are global.

Lemma 2. O ̃ is convex, i.e., O ̃[λq1 + (1 − λ)q2] ≤ λO ̃[q1] + (1 − λ)O ̃[q2].

Proof. O ̃[λq1 + (1 − λ)q2] = min

ρ:λq1+(1−λ)q2=qρ

hOiρ ≤ min

ρ1,ρ2:q1=qρ1

,q2=qρ2

hOiλρ1+(1−λ)ρ2=

min

ρ1,ρ2:q1=qρ1

,q2=qρ2

(λ hOiρ1+ (1 − λ)hOiρ2) = λO ̃[q1] + (1 − λ)O ̃[q2].

Since Omin = hOiρmin ≤ hOiρfor all probability distributions ρ, the global minima of O ̃

coincide with the minima of O, i.e., O ̃[qρmin] = Omin ≤ O ̃[q] for all non-negative

distributions q. Thus it is possible to arrive at the minimum of O by varying q and we have

established the following important theorem.

4
Theorem 1. For a bounded property O that attains its optimum, there exists a convex

functional of non-negative distributions on R

3

that optimizes to a global optimum of O.

This result explains the successes of LCAP and VP-DFT as well as the discrete

optimization approaches. It also generalizes the recent observation that optimization of

molecular properties is considerably more successful than could be expected considering

the vastness of the search space (20). Optimization in convex spaces is fairly efficient, and

the reduction of the optimization space to distributions in only three dimensions instead of

infinitely many is a considerable simplification.

Knowing the distribution without means of interpretation is not particularly useful. But if

the global minimum of O is non-degenerate, then by lemma 1, qmin corresponds to a single

molecule or molecular system. This molecular system can be uniquely identified.

Corollary 1.1. qopt can be uniquely mapped to a composition and conformation.

Proof. Let qopt(R) = P

i

qiδ(R − Ri), then the configuration is the strictly descendingly

ordered sequence of ((qi, Ri))i.

In general applications, boundary conditions of physical viability need to be met. The

introduction of such constraints does not violate the convexity arguments put forth earlier,

since the subspace of probability distributions that are zero whenever the constraint is

violated remains a convex set. Hence, we define the notion of physical distributions for

common requirements.

Definition 6. A distribution P : γ

−1

(c) → R for some composition c = ((qi, Ri))i is

considered physical if P

i



∂E(x)

∂Ri

2

= 0 and Q`(c)

i=1 η0(det Ji) > 0 for all x ∈ γ

−1

(c), where E

denotes the energy and Ji is the ith principal minor of the Hessian of E with respect to the

coordinates Ri, and η0 is the Heaviside step-function with value 0 at 0. A distribution on Γ

is considered physical if every Pi is physical. A distribution q on R

3

is considered

physical if q = qρ for some physical distribution ρ on Γ.

The energy as well as its derivatives are properties that are available over all of

configurational space. Hence, they can be described as convex functionals of the charge

distribution. Using Lagrange multipliers, it is therefore possible to incorporate physicality

constraints using these respective functionals. Let O : Γ → R be a property of interest for

optimization bounded from below (above). In this case, the constrained functional becomes

L ̃ ≡ hLiρ =hOiρ + λghGiρ + λhhJiρ, (2)

where G is the gradient condition and J is the condition of positive-definiteness of the

Hessian, with Lagrange multipliers λg and λh, respectively.

5
3. Approximating O ̃

Since the exact functionals are in most cases unknown, it is necessary to approximate O ̃. It

is desirable that the exact functional can be systematically approximated with decreasing

error. The following lemma proves useful in characterizing appropriate substitutes, as it

obviates the need to be exact on anything but actually physical compounds.

Lemma 3. If a functional W of charge distributions dominates O ̃ from above (below) and

W[q] = min

{c∈Γ:qc=q}

O(c) for all configurationally representable distributions q, then W

optimizes to the same nuclear charge distribution as O ̃.

Proof. W[q] ≥ O ̃[q] ≥ Omin (domination) and W[qmin] = O ̃[qmin] = Omin, because qmin is a

configurationally representable distribution.

We exemplify the use of lemma 3 in the development of an approximate operator to

minimize a non-degenerate electronic property O, e.g., the inverse of the dipole moment,

for neutral species in the ground state using only the nuclear charge distribution. For any

configuration c ∈ Γ, the property O(c) takes the value Dψ





Oˆ





 ψ

E

, where Oˆ is the

corresponding operator on the space of wavefunctions Wkck1 with kck1 electrons and

ψ ∈ Wkck1is the normalized ground-state wavefunction of c. We define W such that it

interpolates linearly between species with integer-valued numbers of electrons:

W[q] = μ1

D

ψ1





Oˆ





 ψ1

E

+ μ2

D

ψ2





Oˆ





 ψ2

E

+ λgG[q, ψ1, ψ2] + λhJ[q, ψ1, ψ2] : (3)

E



bqc

q

q, ψ1



= min

{ψ3∈Wbqc}

E



bqc

q

q, ψ3



(4)

E



dqe

q

q, ψ2



= min

{ψ4∈Wdqe}

E



dqe

q

q, ψ4



(5)

μ1 bqc + μ2 dqe = q, μ1 + μ2 = 1 (6)

The three constraints under consideration in this example are as follows:

1. The ground-state condition stated by equations 4 and 5 and energy functional E

2. The gradient condition G

3. The Hessian positivity condition J

We delay the discussion of the physicality conditions G and J to section 4 and focus on the

general properties of W.

6
In order to apply lemma 3, it is necessary to establish that W dominates O ̃ and coincides

with O ̃ on configurationally representable q. The ground-state condition obeys the

conventional definition:

Hˆ [q] = X

dqe

i=1

−

1

2

∇

2

i −

Z

q(R)

kri − Rk

dR +

X

i<j

1

rij

, (7)

Enuc[q] = Z

Q(X, Y )

kX − Y k

dXdY, (8)

Q(X, Y ) = 1

2

q(X)q(Y ) (1 − δ(Y − X)), (9)

E[q, ψ] = ψ









H



bqc

q

q









ψ



/ hψ|ψi + Enuc 

bqc

q

q



, (10)

E[q, ψ] = ψ









H



dqe

q

q









ψ



/ hψ|ψi + Enuc 

dqe

q

q



, (11)

E[q, ψ1, ψ2] = μ1E[q, ψ1] + μ2E[q, ψ2], (12)

where μ1, μ2 are as before. The Lagrange multipliers enforcing the gradient and Hessian

conditions guarantee that unphysical q are mapped to infinity as they are for O ̃. Hence,

only the behavior of W on physical q is of interest. If q is physical and configurationally

representable, Hˆ [q] and Enuc are precisely the Hamiltonian and nuclear repulsion of the

corresponding configuration c. Therefore, W[q] = O(c) for physical, configurationally

representable q, fulfilling the second requirement of lemma 3.

It remains to be shown that W dominates O ̃ for physical, not configurationally

representable, distributions. The linear-interpolative construction of W with respect to q

reduces the discussion to arguments on integer-valued q only. Without loss of generality, let

q be an admixture of two distinguishable physical, configurations x1, x2 ∈ Γ with

kγ(x1/2)k1 = q and ground-state wave functions ψx1/2. Choose λ ∈ [0, 1], then

q := λqx1 + (1 − λ)qx2 and E[qxi, ψj ] > E[qxi, ψxi], i ∈ {1, 2}, j 6= i, since ψj cannot be the

ground state simultaneously for both distributions. Therefore, the gradient condition is

violated and the Lagrange multiplier pushes the value of W to infinity. Otherwise,

W[q] = O(x1/2) ≥ O ̃[q]. Hence, the first requirement for lemma 3 is fulfilled as well. Since

Omin coincides with a configurationally representable qmin, W attains Omin for qmin, and

therefore, W converges upon minimization to the same composition and conformation as

the general ensemble method does.

4. Enforcing Physicality

The gradient and Hessian conditions will be developed starting from a configurationally

representable distribution. First, we look at the exact gradient functional G before

7
developing an approximate G ̃. It is easily discernable that

G[q] := min

ρ:qρ=q

Z

ρ(x)

X

`(x)

j=1

k∇jEgs(x)k

2

dμ(Γ), (13)

where we have used Egs(x) to denote the ground-state energy and ∇j is the gradient of the

jth nuclear position and is non-negative and zero only if ρ is physical. Given a

configuration, the gradients are derived by perturbing infinitesimally the charge

distribution at a specific point, i.e., δ(R − Ri)

qi

kX−Rk

7→ δ(R − Ri − dRi)

qi

kX−Rk

. The

gradient of the electron-nucleus energy is then taken with respect to the direction dRi.

Hence, the gradient is a small variation of the nuclear charge density. Therefore,

d

∗∇iEgs(x) = lim→0

Z

R3

δEgs[qδΓ,x]

δqδΓ,x

(Y )

qi(δ(Y − Ri + d) − δ(Y − Ri))



dY (14)

=dq(Ri)∇

δEgs[qδΓ,x]

δqδΓ,x

(Ri) = d

∗∇q(Ri)

δEgs[qδΓ,x]

δqδΓ,x

(Ri), (15)

where d is a direction in R

3

. Thus, we can rewrite equation 13 as

G[q] = min

ρ:qρ=q

Z

ρ(x)

Z 











q∇

δEgs[qδΓ,x]

δqδΓ,x













2

dμ(R

3

)dμ(Γ) (16)

= min

ρ:qρ=q

Z

ρ(x)

Z 











∇q

δEgs[qδΓ,x]

δqδΓ,x













2

dμ(R

3

)dμ(Γ). (17)

Two simple approximations given wavefunctions ψ1/ψ2 derive then by dropping all

references to Γ and using the cumulative nuclear charge distribution,

G ̃

1[q, ψ1, ψ2] = Z 











q∇

δE[q, ψ1, ψ2]

δq













2

dμ(R

3

), (18)

G ̃

2[q, ψ1, ψ2] = Z 











∇q

δE[q, ψ1, ψ2]

δq













2

dμ(R

3

), (19)

where we use the linear interpolative form of the energy as used in equation 12. Since these

expressions are non-negative and exact for configurationally representable nuclear charge

distributions, G ̃

1/2 = 0 are good constraints. To satisfy lemma 3, the question remains

whether G ̃

1/2 may be zero for unphysical, non-configurationally representable nuclear

charge distributions. We may restrict the discussion to integer-valued q only. Therefore,

the necessary conditions for fulfilling G1 with the ground-state wave-function φ,

G1[q, φ, φ] = 0

⇒q(X) = 0 or ∇

δE[q, φ]

δq = 0 a.e.,

(20)

may be considered under constraint of constant q. We first show that G1 = 0 is only given

for countable sums of Dirac distributions. Then we show that these are composites of

8
physical distributions. If the set N = R

3 \ {X ∈ R3

: q(X) = 0} has non-zero measure, then

δE[q,φ]

δq is a constant on the completion of N, which contradicts that E[q, ψ, ψ] is a quadratic

functional of q. Therefore, N must be a set of zero measure and, hence, q is a series of

Dirac distributions.

As shown, G1 is related to the norm of the first variation of the energy with respect to the

nuclear charge distribution. If we assume without loss of generality that this distribution

has integer charge, is not configurationally representable, and is a superposition of two

distinct configurationally representable distributions, then the energy functional as a

function of the convex parameter μ1(μ2 = 1 − μ1), as defined before, is a concave

functional. It follows that the gradient condition cannot simultaneously hold with the

Hessian condition for such a distribution.

The Hessian condition J can be derived analogously to G. The Hessian analogue

J[q] = min

ρ:qρ=q

Z

ρ(x)

Z

qx(X)qx(Y )S



∇X∇∗

Y

δ

2E[q, ψ, ψ]

δ

2q



dXdY dμ(Γ) (21)

J1[q] = Zq(X)q(Y )S



∇X∇

∗

Y

δ

2E[q, ψ, ψ]

δ

2q



dXdY (22)

where S(M) := P3

i=1 η0(− det Mi), Mi

is the ith minor of a 3 × 3-matrix M, and η0 is the

Heaviside function, and is positive semi-definite for a configurationally representable energy

minima. Extending finite-dimensional criteria to bilinear forms as in equation 21 and

invoking the gradient condition renders the result that the Hessian matrix for the

configurationally representable distribution q is only non-negative semi-definite if J = 0.

We have thus demonstrated the equivalence of using G1 and J1 to the gradient and Hessian

conditions for use in constrained optimizations on physical distributions.

5. Conclusions

We have laid out a general framework for the optimization of properties of chemical

compounds using only nuclear charge distributions. The unknown exact functional is

convex, which can be exploited for efficient optimization in discrete as well as continuous

approaches. Furthermore, the convexity remains even for constrained problems. In absence

of the exact functional, we have derived an approximate functional for the optimization of

an electronic property under the constraint of physical viability. Ongoing efforts to develop

general purpose electronic basis sets, e.g., wavelets (22–24) or finite element methods

(25–27) open up the potential to use these same bases for efficient, systematic explorations

of chemical space via nuclear charge distributions.

9
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