Mulliken Charges

How do we define the charge of a single atom within a molecule? This seemingly simple question opens a Pandora's box of quantum mechanical complexities. Electrons are not fixed points but exist in diffuse, shared clouds called molecular orbitals, making any division of charge between atoms an act of accounting rather than a direct measurement. This article delves into Mulliken population analysis, one of the earliest and most conceptually simple schemes devised to solve this problem. By exploring its elegant logic—and its spectacular failures—we can gain profound insight into the very nature of chemical bonding and the distinction between a useful model and physical reality.

This article will first unpack the core ideas in "Principles and Mechanisms," detailing how Mulliken charges are calculated through the LCAO method and the democratic split of the overlap population, while also exposing the method's fatal flaws, such as its dependence on the basis set. Subsequently, the "Applications and Interdisciplinary Connections" chapter will explore the practical uses of this model, from sketching bond character to illuminating photochemical processes, and explain why, despite its utility, it must be used with significant caution, especially when compared to more physically robust modern methods.

How do we talk about the charge on an atom inside a molecule? It seems like a simple question. We learn in chemistry that in a water molecule, the oxygen is a bit negative and the hydrogens are a bit positive. This simple picture helps us understand everything from why water is a liquid to how enzymes work. But what does it really mean for an atom to "have" a charge?

In the quantum world, electrons aren't tiny billiard balls that belong to one atom or another. They are spread out in diffuse clouds of probability called ​​molecular orbitals​​, which can span the entire molecule. So, if we want to assign a charge to an atom, we are forced to play a game of accounting. We must invent a rule, a scheme, to divide up the molecule's total electron cloud and assign portions of it to each atomic nucleus. Mulliken population analysis is one of the oldest and simplest of these schemes, and by understanding its elegant logic—and its spectacular failures—we can learn a great deal about the nature of chemical bonds.

Imagine a simple diatomic molecule, say AB, where the bond is formed by two electrons occupying a single molecular orbital, $\psi$. In the popular ​​Linear Combination of Atomic Orbitals (LCAO)​​ picture, we imagine this molecular orbital is constructed by mixing together building blocks from each atom—the atomic orbitals, $\phi_A$ and $\phi_B$. So, we write:

The coefficients $c_A$ and $c_B$ tell us the "amount" of each atomic orbital in the mix. The total number of electrons in this orbital is two. The probability of finding an electron at any point in space is proportional to $|\psi|^2$. If we expand this, we get:

This equation reveals three parts to the electron distribution. The term $c_A^2 |\phi_A|^2$ represents electron density that looks like it belongs to atom A. The term $c_B^2 |\phi_B|^2$ is the part that looks like it belongs to atom B. But what about that third term, $2 c_A c_B \phi_A \phi_B$? This is the ​​overlap population​​. It represents the electron density that isn't purely A-like or B-like, but exists in the bonding region shared between them.

This is where Robert S. Mulliken's simple and beautifully democratic idea comes in. He proposed a rule: for the part of the electron density that is clearly associated with atom A's orbitals, we assign it all to A. For the part associated with B's orbitals, we assign it all to B. For the overlap population, which is shared, we simply split it down the middle: 50% for A and 50% for B.

This leads to a wonderfully simple recipe for the total number of electrons assigned to an atom, its ​​gross atomic population​​. Let's call it $N_A$. The charge on the atom, $q_A$, is then just the charge of its nucleus, $Z_A$ (the number of protons), minus the number of electrons we've assigned to it.

In the language of matrices that computational chemists use, this entire recipe can be summarized in a single, elegant line. The gross population on atom A is given by summing up specific elements of the product of the ​​density matrix​​ ($P$) and the ​​overlap matrix​​ ($S$). For a simple diatomic molecule with one basis function on each atom, this leads to a clear expression for the charge on atom A:

Here, the expression in the parenthesis is the gross electron population on atom A ($N_A$). The term $P_{AB}S_{AB}$ represents A's half of the shared overlap population. It's a neat and tidy accounting system.

Does this simple scheme work? Sometimes, surprisingly well! Consider the series of molecules $CH_4$, $NH_3$, $H_2O$, and $HF$. As we move from carbon to fluorine, the central atom becomes progressively more electronegative—it gets better at pulling electrons towards itself. We would expect the hydrogen atoms to become more and more positively charged along this series. A Mulliken analysis predicts exactly this trend. The simple 50/50 split, while arbitrary, is consistent enough to capture this fundamental chemical behavior.

But this is where the story gets interesting. The Mulliken recipe can give answers that defy chemical intuition, a classic example being lithium hydride, LiH. Lithium is a highly electropositive element, and hydrogen is more electronegative, so we unequivocally expect the bond to be polarized as $\text{Li}^{\delta+}\text{H}^{\delta-}$. And indeed, many simple calculations give this intuitive result. However, if one includes diffuse functions in the basis set, particularly on the hydrogen atom, Mulliken analysis can paradoxically calculate a negative charge on lithium and a positive charge on hydrogen. Is the calculation wrong? No, the accounting is perfectly correct according to its own rules. The diffuse basis function on hydrogen is so large that it effectively blankets the lithium atom. Mulliken's protocol, blindly following its 50/50 rule, assigns a large portion of the electron density to hydrogen's basis functions, even the density that is physically near the lithium nucleus. The result is an artifact: it reveals that Mulliken's scheme is more a statement about the mathematical properties of the basis functions than the physical reality of the molecule.

This brings us to the central, fatal flaw of Mulliken's beautiful idea: its results depend dramatically on the "atomic orbitals" we choose as our building blocks, a choice known as the ​​basis set​​. Is the charge on an atom a true physical property, like mass or temperature? If it were, any two sensible scientists using different but equally good tools to measure it should get the same answer.

This is not the case for atomic charges. Methods that derive charges by fitting them to reproduce the molecule's external ​​electrostatic potential (ESP)​​ are often considered more physically robust. Unsurprisingly, ESP charges can differ significantly from Mulliken charges, leading to different predictions for physical properties like the molecular dipole moment. This tells us that "atomic charge" is not one unique number; it's a model, and the answer you get depends on the model you use.

Mulliken's model is particularly fragile. Its Achilles' heel is that 50/50 split of the overlap population. The size of this overlap depends sensitively on how spatially extended our chosen basis functions are. In modern quantum chemistry, we often use very flexible basis sets that include ​​diffuse functions​​—big, fluffy orbitals that spread far out from the nucleus. This is where things fall apart.

Imagine trying to describe a location using two street signs that are almost on top of each other. It's redundant, and any small error in reading one sign might require a huge, compensating error in reading the other to get the right final position. Adding diffuse functions to a basis set does something similar. It can create a situation of ​​near-linear dependence​​, where different basis functions on different atoms become nearly identical in the space they describe. Mathematically, this is flagged by the overlap matrix $S$ having a very small eigenvalue.

In this situation, the calculation can still produce a good overall energy and electron density, but the individual LCAO coefficients for the redundant orbitals can become wildly large with opposite signs. The Mulliken formula, which relies on these individual components, goes haywire. The "overlap population" it tries to divide can become nonsensically large, negative, or positive. The resulting charges become meaningless noise. This instability is a core feature of the method, not a bug in the computer program. This is why a common saying among computational chemists is that the Löwdin charge (a related but more stable scheme) is a property of the molecule and the basis set, but the Mulliken charge is a property of the molecule, the basis set, and the alignment of the planets. A more sober analysis reveals that both are basis-set dependent, but Mulliken's method is pathologically so.

Several practical warning signs can tell you that your Mulliken charges are unreliable:

• Seeing individual basis function populations that are negative or greater than 2.
• Observing that the charges change dramatically when a slightly different, but still reasonable, basis set is used.
• Finding that adding a single, non-physical "ghost" function far away from the molecule significantly changes the charges on the atoms. A robust property shouldn't be affected by something happening far away.
• Even the orientation of the molecule can affect the charges if the basis functions are defined relative to a fixed external coordinate system, another sign that we are calculating an artifact of our method, not an intrinsic property of the molecule.

What if we could use a "perfect" or ​​complete basis set​​, one with infinite flexibility? Surely then the Mulliken charge would converge to a true, physical value? The answer is a resounding and definitive no. In the complete basis set limit, the problem of linear dependence becomes infinitely bad. The Mulliken charges do not converge to a unique value; they can be made to be almost anything you want depending on the precise way you approach the limit. It is a path to nowhere.

The journey through Mulliken's world is a fascinating one. It begins with a simple, intuitive idea for dividing up a molecule's electrons. It proves useful for understanding simple chemical trends, but it quickly leads to paradoxes and counter-intuitive results. Finally, as we push it with the powerful and flexible tools of modern computation, the entire structure collapses, revealing itself to be a mathematical artifact. And in that collapse, we learn a profound lesson: some simple, intuitive questions we like to ask in science, like "what is the charge on this atom?", may not have a simple, unique answer in the rich and interconnected quantum reality.

Now that we have explored the machinery behind Mulliken's method for assigning charges to atoms, a natural and pressing question arises: What is it good for? After all, a scientific tool is only as valuable as the insights it provides and the problems it helps us solve. As we shall see, Mulliken charges are a fascinating case study in the power and peril of scientific approximation. They are like a brilliant cartoon sketch of a complex reality—not a photograph, to be sure, but capable of capturing essential character and telling a compelling story, provided we remember we are looking at a sketch.

One of the most fundamental ideas in chemistry is the distinction between an ionic bond, where an electron is fully transferred from one atom to another, and a covalent bond, where electrons are shared between atoms. In reality, most bonds lie somewhere on a spectrum between these two extremes. How can we get a feel for where a particular bond falls on this spectrum?

Mulliken analysis offers a beautifully simple, if somewhat naive, answer. As we've learned, the method works by partitioning the total electron population. It assigns some electrons exclusively to one atom or another, but it also finds a certain number of electrons in the "overlap" region, the no-man's-land of shared space between atoms. This "overlap population" is the very essence of covalency. Mulliken's simple prescription is to split this shared population right down the middle, giving half to each atom involved in the bond.

This simple rule has a powerful consequence. If we analyze a molecule like sodium chloride, where the bond is almost purely ionic ($\text{Na}^+\text{Cl}^-$), the calculation finds very little shared electron density. The Mulliken charges come out very close to the formal integer charges of $+1$ and $-1$. But if we look at a molecule like nitric oxide (NO), the story is different. Oxygen is more electronegative than nitrogen, so it certainly pulls electron density toward itself. The Mulliken analysis confirms this, giving oxygen a negative charge and nitrogen a positive one. However, the charges are nowhere near $-1$ and $+1$. This tells us that a significant portion of the electron density was found in that shared, overlap region and was split between the two atoms. The deviation from integer charges becomes a direct, quantitative measure of the bond's covalent character. It provides a numerical language for what our chemical intuition tells us: the bond is polar, but also strongly covalent.

The utility of this "electron accounting" goes far beyond static pictures of ground-state molecules. Chemistry is about change, and one of the most dramatic changes a molecule can undergo is absorbing a photon of light. When this happens, an electron is promoted to a higher energy level, and the entire electronic landscape of the molecule can shift in an instant.

Consider a specially designed molecule like dimethylaminobenzonitrile (DMABN), a classic subject in the field of photochemistry. This molecule has two distinct parts: an electron-rich "donor" group and an electron-accepting "acceptor" group. When it absorbs ultraviolet light, an electron effectively leaps from the donor to the acceptor in a process called intramolecular charge transfer (ICT). This phenomenon is not just a curiosity; it is the fundamental principle behind technologies like organic light-emitting diodes (OLEDs) and some types of solar cells.

Mulliken charges provide a way to visualize and quantify this electronic dance. By performing one calculation for the molecule in its ground state and another for its excited state, we can obtain two sets of Mulliken charges. By summing the charges on the atoms in the donor fragment before and after excitation, we can calculate precisely how much charge has fled. The result is a single number, expressed in fractions of an elementary charge, that characterizes the efficiency of the charge transfer. This simple numerical sketch helps chemists and materials scientists understand and design molecules with specific photophysical properties.

Let us now push our simple tool into a truly exotic realm: the world of heavy elements, where electrons move so fast that Einstein's theory of special relativity becomes important. For an atom like thallium (Tl), which is far down the periodic table, the immense positive charge of the nucleus ($Z=81$) accelerates the innermost electrons to a significant fraction of the speed of light. According to relativity, this makes them behave as if they are heavier, causing their orbits (specifically the $s$ orbitals) to contract and fall closer to the nucleus.

This has a remarkable chemical consequence known as the "inert pair effect." These relativistically stabilized $s$ electrons become chemically "shy," reluctant to participate in bonding. How could our simple Mulliken charge model possibly detect such a profound physical effect?

The answer is stunning. Imagine we perform two separate quantum chemical calculations on thallium(I) hydride (TlH). The first is a "normal," non-relativistic calculation. The second is a scalar-relativistic calculation that accounts for these effects. We then compute the Mulliken charges for both cases. The non-relativistic calculation predicts a certain positive charge on the thallium atom, indicating it has pulled some amount of electron density from hydrogen. But the relativistic calculation tells a different story: the positive charge on thallium is significantly smaller. The Mulliken charge has numerically captured the inert pair effect! Because the relativistic calculation correctly makes the thallium $s$ electrons more inert, they are less able to pull electron density from the hydrogen atom through bonding. The molecule becomes less polar, and the Mulliken charge faithfully reports this change. It is a powerful demonstration of how a simple interpretive tool can provide a window into the deepest principles of physics governing chemical bonds.

By now, Mulliken charges might seem like a magical panacea. They are computationally cheap and appear to grant us deep chemical insights. It is at this point that we must, in the spirit of good science, discuss the catch. And it is a very big catch.

The trouble begins when we try to use Mulliken charges not for qualitative sketches, but as quantitative parameters in other models, particularly the "force fields" used for large-scale molecular dynamics (MD) simulations of proteins, liquids, and materials. These simulations model atoms as balls and springs, and a crucial component of the model is the electrostatic interaction, which is calculated using a fixed partial charge assigned to each atom.

It seems temptingly straightforward to calculate Mulliken charges and plug them directly into an MD simulation. This, however, is a classic trap. The first problem is that Mulliken charges are notoriously sensitive to the "basis set"—the set of mathematical functions used to approximate atomic orbitals in the quantum calculation. Adding more flexible functions, particularly diffuse ones that spread far from the nucleus, can cause the arbitrary "split the middle" rule to behave pathologically, yielding charges that swing wildly and are physically meaningless. A robust physical quantity should not depend so sensitively on the arbitrary details of our mathematical setup. It's like measuring a coastline: the answer you get depends on the length of your ruler.

The second, more fundamental problem lies in the goal itself. The purpose of a force field charge is to reproduce the electrostatic potential outside the molecule—the field that a neighboring water molecule or drug would actually "feel." Mulliken analysis, by contrast, is an inward-looking scheme that partitions the density inside the molecule. There is no guarantee that its partitioning will reproduce the correct external field.

For this reason, a different class of charges, known as Electrostatic Potential (ESP)-derived charges, is now the gold standard for force field parameterization. The ESP approach is physically much more sound: it first calculates the "true" quantum mechanical electrostatic potential around the molecule and then finds the set of atom-centered point charges that best reproduces this external field.

The choice is not merely academic; it has real, measurable consequences. If you try to calculate a thermodynamic property like the hydration free energy of methanol—a measure of how well it dissolves in water—you will get a significantly different (and more accurate) answer using ESP charges than with Mulliken charges. The ESP charges, which are typically larger for polar groups to implicitly mimic electronic polarization in the condensed phase, lead to a more realistic description of the strong attractive forces between methanol and water. Similarly, in hybrid QM/MM simulations used to study enzymes, the choice of charge scheme for the atoms at the boundary between the quantum and classical regions can drastically alter the calculated interaction energies, potentially changing the entire conclusion of the simulation [@problemid:2457580].

So, where does this leave us? The story of Mulliken charges is a profound lesson in the nature of scientific modeling. They are not a fundamental physical observable. They are an interpretation, a projection of a complex quantum reality onto a simple, chemically intuitive picture. This instability and dependence on arbitrary choices make them a poor foundation for quantitative, predictive models and an unreliable metric for technical tasks like judging the convergence of a quantum calculation.

Yet, they remain a valuable tool for building qualitative understanding. When used with caution, they help us translate the dense mathematics of quantum mechanics into narratives about chemical bonding, charge transfer, and even relativistic effects. They provide a first, quick look—a sketch. Other, more sophisticated methods like Hirshfeld analysis, Natural Bond Orbital (NBO) analysis, or Quantum Theory of Atoms in Molecules (QTAIM) provide different sketches, each with its own strengths and weaknesses. The true art of the computational scientist is not just to generate numbers, but to understand the story each method is telling, to recognize its inherent biases, and to choose the right tool—or the right collection of tools—for the question at hand. The journey of understanding Mulliken charges is, in essence, a journey into the art of scientific judgment itself.