The solution: It does exist but can not be written in closed form

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First there is some good news: The solution to the many-body problem exists. You might laugh at this conclusion (otherwise we ourselves would perhaps not exist...) but it is mathematically highly non-trivial. Anyway, the bad news is: It can not be written in "closed form". What does "closed form" mean? Let us assume the solution is a function of x: f (x). A closed form would be for example f (x) = sin(x). Non-closed form means that the function f (x) must be expanded in an infinite sum of other functions g_i(x) of closed form:

f (x) = $$\sum_{{i=0}}^{\infty}$$c_ig_i(x) = c₀g₀(x) + c₁g₁(x) + ....

Excursion: Expansions
This function expansion thing may seem somewhat unfamiliar. Let us simplify the problem and consider numbers not functions. Think of the number $\pi$ = 3.14159.... $\pi$ is an irrational number. An irrational number can not be written as a fractional number: $\pi$$\ne$${p\over q}$. Instead $\pi$ must be expanded in an infinite series. For example it is

$$\pi$$ = 4$$\sum_{{i=0}}^{\infty}$$$${(-1)^i\over 2i+1}$$ = $${4\over 1}$$ - $${4\over 3}$$ + $${4\over 5}$$ -...

Let us try the first few terms:

$$\pi \simeq$$ 4 = 4

$$\pi \simeq {4\over 3}$$

$$\pi \simeq {4\over 3} {4\over 5}$$

$$\vdots$$

$$\pi \simeq {4\over 3} {4\over 5} {4\over 21}$$

We believe (it can be proven): This series will end up with $\pi$. However, this series does not converge very fast. There are better series. For example:

$${\pi^4 \over 90}$$ = $$\sum_{{i=1}}^{\infty}$$$${1\over i^4}$$ = $${1\over 1^2}$$ + $${1\over 2^2}$$ + $${1\over 3^2}$$ +...

It is ${\pi^4 \over 90}$ = 1.0823232.... Let us try the first few terms again:

$${\pi^4 \over 90} \simeq$$ 1 = 1

$${\pi^4 \over 90} \simeq {1\over2^4}$$

$${\pi^4 \over 90} \simeq {1\over2^4} {1\over3^4}$$

$$\vdots$$

$${\pi^4 \over 90} \simeq {1\over2^4} {1\over3^4} {1\over 10^4}$$

Now we have got already 4 digits after 10 expansion terms compared to only 1 correct digit for the upper expansion. Obviously the latter expansion converges more rapidly.

Back to the real problem
Now we turn back to the many-body problem where we look for a function solving a partial differential equation. In principal the situation is fairly analogous. There may be fast and slow convergent expansions. Of course one prefers fast convergent expansions. The construction of these fast converging series is the main goal of many-body physics and theoretical chemistry.

Goals beside "exactness"
So far we know: The solution exists but it can never be written exactly since we can not set up a really infinite series. In fact, compared to the simple expansion of $\pi$ into a series the computational cost of the many-body function expansion will explode with every significant digit. So there should be additional goals for a good wavefunction besides exactness. And there are: One may show that the exact solution has certain properties. Obviously, it would be nice for an approximate wavefunction to reproduce as many of those properties of the exact solution as possible although it is itself not exact. An example for such a property would be that the calculated energy should be either always below or above the (unknown) exact energy. In the context of the wavefunction this property is called "variational". Looking at the two series expansions of $\pi$ we see that the first one fails to meet this criterion since it is once below and once above changing with every term in the expansion. The second expansion however does meet this criterion: The estimate for $\pi$ is always below the exact one. Another such property (so called "size extensivity") of the exact wavefunction is that the energy calculated from it is proportional to the size of the system (if the system is sufficiently large). Size extensivity turns out to be much more important for the accuracy of a calculation than variationality. It supports a fast convergent expansion of the exact wavefunction.

Configuration interaction (CI) methods
An example for a wavefunction method meeting the variationality condition is the so called configuration interaction approach. It is conceptually very simple. The major advantage of this approach is that it is generally applicable (multi-reference configuration interaction, MRCI). However, convergence to the exact result is rather slow since CI does not meet the size extensivity condition.

Coupled cluster (CC) methods
Size extensivity sounds like a trivial demand but it turns out that it is not. Many methods (among them CI) fail with respect to this demand. However, there are methods which meet this requirement. One of those is the so called coupled cluster method. The coupled cluster method was invented around 1958 and the guys who came up with it predicted it to be "of no practical use" since the resulting equations were considered to be too complicated. Nevertheless, time has changed and today the coupled cluster approach is one of most popular and efficient methods to account for the many-body problem. However, there is a severe limitation of the coupled cluster method: It is not applicable to all types of problems since it requires a certain structure of the wavefunction which is often not met. Therefore lots of theoreticians (including myself) worked and still work on the generalization (multi-reference coupled-cluster, MRCC) of the coupled cluster approach from 1958 but nobody came up with the ultimate solution so far. Unfortunately, this is very difficult problem.