Complex harmonics against real ones

Solving the hydrogen atom gives a set of complex functions labelled by $n$, $l$ and $m$. Every chemistry textbook then draws $p_x$, $p_y$ and $p_z$, which are none of them.

The substitution is legitimate, universal, and made almost always without comment.

What the solutions actually are #

The angular part of a hydrogenic wavefunction is a spherical harmonic $Y_l^m(\theta, \phi)$, and it carries a factor $e^{im\phi}$.

For $l = 1$ the three solutions have $m = -1, 0, +1$. The $m = 0$ one is real and is $p_z$. The other two are genuinely complex: $Y_1^{\pm1} \propto \sin\theta\, e^{\pm i\phi}$, and neither has a shape that can be drawn as a signed lobe pattern, because its value at a point is a complex number.

What is drawn instead are the combinations

$$p_x \propto \tfrac{1}{\sqrt2}(Y_1^{-1} - Y_1^{+1}), \qquad p_y \propto \tfrac{i}{\sqrt2}(Y_1^{-1} + Y_1^{+1}),$$

which are real, are orthonormal, and span exactly the same two-dimensional space.

The same substitution is made for d and f orbitals. The five familiar d shapes — $d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$ and $d_{z^2}$ — are real combinations of the five complex harmonics, with only $d_{z^2}$ ($m=0$) being a solution as it stands.

Why both sets are equally correct #

Because the transformation between them is unitary, and a unitary transformation of a degenerate set changes nothing observable.

The three $l=1$ functions are degenerate: they have the same energy, so any combination of them is also an eigenfunction of the Hamiltonian with that energy. The real set and the complex set are two orthonormal bases for one three-dimensional space, and every quantity that depends only on the space — the total density of a filled subshell, the total energy, the number of orbitals — is the same in both.

This is the same argument the localisation transformation makes for methane’s bonding orbitals, arriving three chapters earlier and passing unremarked.

What each set is an eigenfunction of #

Both sets are eigenfunctions of the Hamiltonian and of $\hat{L}^2$. They differ in one operator.

The complex set are eigenfunctions of $\hat{L}_z$, the component of angular momentum along the chosen axis, with eigenvalues $m\hbar$. That is what the label $m$ means.

The real set are not. $p_x$ is a superposition of $m = +1$ and $m = -1$, so a measurement of $L_z$ on an electron in $p_x$ returns $+\hbar$ or $-\hbar$ with equal probability and never returns the average.

So the two sets answer different questions. If the question is about angular momentum, the complex set is the natural basis. If the question is about direction in a molecule, the real set is, because $p_x$ points along $x$ and $Y_1^{+1}$ points nowhere in particular.

Chemistry asks about direction almost always, which is why chemistry uses the real set almost always.

When the distinction bites #

Three cases, and they are the whole of the answer to “does this ever matter”.

A magnetic field. The field defines an axis and lifts the degeneracy: states of different $m$ acquire different energies, proportional to $m$. That is the Zeeman effect, and once the degeneracy is lifted the three complex functions are no longer interchangeable with any combination of themselves. The real set stops being a valid basis in the sense that matters — $p_x$ is no longer an eigenstate, because the three states it is built from now have different energies.

That is the sharpest statement available: the freedom to choose a basis exists only within a degenerate set, and a magnetic field destroys the degeneracy.

Angular momentum coupling. Spin–orbit coupling, term symbols, and every selection rule involving $\Delta m$ are statements about $L_z$ eigenvalues. Working them in the real basis is possible and is much harder, because the operator is not diagonal there.

Circular polarisation. A photon carries one unit of angular momentum, so absorbing a circularly polarised photon changes $m$ by a definite amount. Circular dichroism and magnetic circular dichroism are experiments that distinguish $m = +1$ from $m = -1$ directly, and they cannot be described in a basis where those two are mixed.

Outside those three, the choice is free and the real set is more convenient.

The detail almost everybody gets slightly wrong #

The $d_{z^2}$ orbital is not called $d_{z^2}$ by anybody being careful, and the reason is a small instance of this essay’s subject.

Its angular function is proportional to $3z^2 - r^2$, so its proper name is $d_{2z^2 - x^2 - y^2}$, and it is written $d_{z^2}$ purely as an abbreviation.

The reason it looks different from the other four — two lobes and a doughnut, rather than four lobes — is that the five real d functions cannot all be made to look alike. Six combinations of the form $d_{x^2-y^2}$, $d_{y^2-z^2}$, $d_{z^2-x^2}$ and the three $d_{xy}$-type functions would be symmetric, and six is one too many: the three “difference” functions satisfy one linear relation, so only two of them are independent.

So the odd one out among the d orbitals is an artefact of a basis choice, and a different choice of the five real combinations would move the oddity elsewhere. Nothing about the atom singles out $d_{z^2}$.

What was computed, and how #

The real harmonics here are written out explicitly rather than assembled from complex ones, which is a simplification worth declaring: no complex arithmetic appears anywhere in the calculation.

What is checked is that the set is a valid basis. Every orbital’s density integrates to one to a part in $10^6$; every pair of distinct orbitals on the same centre is orthogonal to the same tolerance; and the angular node count is $l$ for every one of them, counted by walking many randomly oriented great circles and taking the largest number of sign changes.

That last check needs its complexity. Walking one great circle works for a p orbital and fails badly for others — a circle lying in the plane containing both nodal planes of a $d_{xy}$ orbital touches them tangentially, the function goes as $\cos^2 t$ along it and never changes sign, and the count comes out zero for an orbital with two nodal surfaces, while the drawing stays correct throughout.

The counter is also handed an orbital claiming the wrong number and required to refuse it, and the orthogonality check is handed an orbital declared orthogonal to itself.

The complex functions are not computed anywhere. Everything on this page about $Y_1^{\pm1}$ is analysis rather than computation, and a figure of a complex harmonic would require a choice of how to display a complex-valued function, which is a different problem from the one this site solves.

The surprise: a picture cannot show which basis it is in #

The observation that ties this essay to the rest of the site is that no orbital picture records its own basis choice.

A drawing of $p_x$ and a drawing of $p_z$ are the same shape in different orientations. Nothing in either says whether the artist chose the real set or happened to be drawing the $m=0$ function. Nothing distinguishes a degenerate set that has been rotated into a convenient orientation from one that was solved for in that orientation.

That is the same blindness an orbital contour has about its level: the picture carries less information than the object, and the missing information is a parameter somebody chose.

The difference is that a contour level can be stated in a caption, and a basis choice largely cannot — there is no short phrase that would distinguish “this is $Y_1^0$” from “this is the real combination oriented along $z$”, and for a filled subshell there would be nothing to distinguish, because a filled subshell has spherical density in either basis.

Which is itself the cleanest demonstration of the essay’s claim. Fill all three p orbitals and the total density is a sphere, in the real basis and in the complex one, identically. The basis has vanished from the observable.

What it costs #

Nothing computationally: the real harmonics are five short expressions.

The cost is in what has to be unlearned later, and it is a real cost paid by every student who meets magnetism.

A first course draws real orbitals, labels them by shape, and never mentions $m$ except as a counter for how many orbitals there are in a subshell. A second course introduces $m$ as an angular momentum quantum number with physical consequences — and the orbitals it applies to are not the ones the first course drew.

The two accounts are consistent and the bridge between them is one sentence about a unitary transformation, which is very rarely said. What gets learned instead is that $m$ labels the three p orbitals, which is wrong in a way that matters exactly when the degeneracy is lifted.

There is a smaller cost that is more often noticed and less important: the habit of saying “the electron is in the $p_x$ orbital” for an atom in free space, where $x$ has been chosen by whoever drew the picture and no direction is physically distinguished at all.

Where the model stops #

Three limits.

Degeneracy is the licence. The freedom to recombine exists only among functions of the same energy. In a molecule, symmetry can split what is degenerate in an atom, and the surviving freedom is only within the sets the molecular group keeps degenerate.

This is a one-electron statement. The argument concerns a set of one-electron functions. In a many-electron atom the orbitals are an approximation, and what is really being recombined is the basis of that approximation.

And a filled subshell hides everything. The whole distinction becomes invisible for a closed shell, which is the case a great deal of chemistry deals with. That is convenient and it means the concept is usually met first in a context where it makes no difference, which is a poor place to learn it.

The one place the real set is genuinely worse #

Everything above has argued that the choice is free within a degenerate set. It is worth closing with the case where the real set is not merely inconvenient but actively misleading, because it is a case a chemist meets.

A transition-metal ion in an octahedral field has its five d orbitals split into a set of three and a set of two, and every textbook draws the five real functions and assigns them to the two sets. That works, and it works because the octahedral group happens to keep those particular combinations apart.

Rotate the field — put the same ion in a tetragonal or trigonal environment — and the correct combinations are different real ones. The five pictures a reader has memorised are correct for one orientation of one field, and the habit of treating them as the five d orbitals survives the change of environment when it should not.

The general principle is the one site symmetry states: which combinations are equivalent is decided by the group of the environment, and a set of pictures fixed once cannot record a group that changes.

The orbital that looks different from the other four #

The same choice has a consequence in the d shell that causes more confusion than the p shell’s, and it is worth disposing of, because it produces a question every student asks and few textbooks answer.

Four of the five real d orbitals look alike: four lobes in a plane, alternating in sign, differing only in orientation. The fifth, conventionally drawn with two lobes along an axis and a ring around the middle, looks like a different kind of object — and it is routinely asked why one of the five is special.

It is not. The five are one set of functions of the same kind, and the odd appearance is a consequence of writing a two-dimensional piece of the set with real functions.

Three of the five — the ones with lobes between the axes — belong together and can be written as three clover shapes with no difficulty. The remaining two form a pair, and any two orthogonal combinations of that pair will do. The conventional choice takes $x^2 - y^2$ for one and, for the other, the combination usually written $z^2$ but which is really $2z^2 - x^2 - y^2$ — a difference of two clover-like functions, which is where the ring comes from.

A different and equally valid choice writes the same pair as $z^2 - x^2$ and $z^2 - y^2$: two functions of identical shape, each looking exactly like the four others, with no ring anywhere and no odd member. That set is not orthogonal in the usual way, which is why it is not the convention — but it demonstrates that the ring is a property of the choice rather than of the shell.

So the answer to why does one d orbital look different is that none of them does. Two of the five are being written as a difference and a sum instead of as two of a kind, and the picture inherits the arithmetic. It is the same lesson this essay makes about the p orbitals, one shell along and with a more visible symptom.

Who found it, and when #

The spherical harmonics predate quantum mechanics by more than a century. Laplace and Legendre developed them in the 1780s for gravitational potential theory, and they arrived in atomic physics as ready-made mathematics.

The real combinations are older than their chemical use too — they are the standard real basis for the same functions, used in geodesy and in the theory of the potential long before anybody drew an orbital.

The convention of drawing the real set in chemistry is essentially Pauling’s, from the 1930s, and it was a good choice for the reason given above: chemistry is about directions in molecules, and the real functions point along directions.

The complex set kept its place in atomic spectroscopy the whole time, which is why a reader moving between a chemistry text and a spectroscopy text meets two apparently different sets of p orbitals with no note that they are the same three functions in different coordinates. Two literatures, one space, and a transformation nobody writes down.

Where to read on #

The object being drawn is what an orbital is.

The general claim about bases is hybrids are a basis, and its numerical demonstration is the localisation transformation.

The count that survives every basis choice is nodes.

And the other parameter an orbital picture fails to record is the contour level.

The habit worth carrying away is a small scepticism about orbital pictures generally. Two parameters have to be chosen before one can be drawn — the contour level and the basis — and a picture records neither. The pictures here state the first in every caption. The second cannot be stated in a caption, which is why it gets an essay.

One consolation for a reader who finds all this unsettling. Every quantity a first course actually computes with orbitals — energies, node counts, densities of filled shells, symmetry labels in a molecule — is basis-independent, so nothing learned in the real basis has to be unlearned. What has to be added is the sentence saying that a basis was chosen, and the three circumstances in which the choice stops being free.

There is one more reason the distinction is worth carrying, and it is about reading rather than about physics. A paper that says an electron is in the $m = +1$ orbital and a paper that says it is in $p_x$ are not disagreeing, are not describing different systems, and are very likely to be describing the same measurement. Recognising that at sight saves more time than any of the three cases above.

The convention chemistry settled on is the right one for chemistry, and it is a convention.

What the pictures here cannot show. Every figure on this page is a real function, because a complex-valued function has no signed-lobe drawing — and the essay is about functions that are complex. What is drawn is one basis for the space; the other basis cannot be drawn in the same idiom at all, and its absence from this page is not an omission but the point.