dustinkaiser.euhttps://dustinkaiser.eu/2026-04-14T00:00:00-04:00Dustin KaiserCoupled Cluster Theory Through the Lens of Cumulants2026-04-14T00:00:00-04:002026-04-14T00:00:00-04:00Dustin Kaisertag:dustinkaiser.eu,2026-04-14:/blog/2026/coupled-cluster-theory-through-the-lens-of-cumulants/<p>The coupled cluster ansatz is really a cumulant expansion in disguise — and that tells us exactly when it should break down.</p><p>One of the most satisfying connections I've found recently is between
coupled cluster (CC) theory and cumulant expansions from probability
theory.</p>
<h2 id="the-exponential-ansatz">The exponential ansatz<a class="headerlink" href="#the-exponential-ansatz" title="Permanent link">¶</a></h2>
<p>The CC wave function is written as:</p>
<div class="math">$$|\Psi\rangle = e^{\hat{T}} |\Phi_0\rangle$$</div>
<p>where <span class="math">\(\hat{T} = \hat{T}_1 + \hat{T}_2 + \cdots\)</span> is the cluster operator.
This exponential structure is not just a convenient parameterization — it's
a <em>cumulant-generating</em> construction.</p>
<h2 id="why-this-matters">Why this matters<a class="headerlink" href="#why-this-matters" title="Permanent link">¶</a></h2>
<p>In probability theory, cumulants <span class="math">\(\kappa_n\)</span> relate to moments through:</p>
<div class="math">$$\ln \mathbb{E}[e^{tX}] = \sum_{n=1}^{\infty} \kappa_n \frac{t^n}{n!}$$</div>
<p>A Gaussian distribution has <span class="math">\(\kappa_1 = \mu\)</span>, <span class="math">\(\kappa_2 = \sigma^2\)</span>, and
<span class="math">\(\kappa_{n \geq 3} = 0\)</span>. CCSD — which truncates at <span class="math">\(\hat{T}_2\)</span> — is making
the same approximation: it assumes the many-body correlation is well
described by its first two cumulants.</p>
<h2 id="a-diagnostic-for-reliability">A diagnostic for reliability<a class="headerlink" href="#a-diagnostic-for-reliability" title="Permanent link">¶</a></h2>
<p>This gives us a clear criterion for when CC should work: the CI coefficient
distribution should be approximately Gaussian. We can check this with the
configuration entropy:</p>
<div class="math">$$S = -\sum_I |c_I|^2 \ln |c_I|^2$$</div>
<p>When <span class="math">\(S/S_{\max} \lesssim 0.3\)</span>, the distribution is dominated by a few
configurations (approximately Gaussian in the cumulant sense), and CC
converges rapidly. When it approaches 1, higher cumulants matter and CC
methods struggle — the onset of strong correlation.</p>
<p>More on this in a future post, where I'll trace the connection through
to Jaynes' maximum entropy framework.</p>
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</script>Welcome2026-04-14T00:00:00-04:002026-04-14T00:00:00-04:00Dustin Kaisertag:dustinkaiser.eu,2026-04-14:/blog/2026/welcome/<p>First post — what this site is about and what to expect.</p><p>This is my place for writing about science, code, and the
ideas that connect them.</p>
<p>I've been working at the intersection of computational science and
software engineering for a while, and I've accumulated enough stray
ideas that they deserve a place to live. Some posts here will be
technical deep dives. Others will be shorter notes on tools, workflows,
or things I've learned the hard way.</p>
<h2 id="what-to-expect">What to expect<a class="headerlink" href="#what-to-expect" title="Permanent link">¶</a></h2>
<ul>
<li><strong>Science posts</strong>: explorations of quantum chemistry, information theory,
and statistical mechanics — the kind of material that lives between
textbook chapters.</li>
<li><strong>Code posts</strong>: practical write-ups on Python, scientific computing,
infrastructure, and tooling.</li>
<li><strong>Notes</strong>: shorter pieces on things I found useful or surprising.</li>
</ul>
<p>If any of that sounds interesting, stick around.</p>