```python
from sympy import init_session
init_session()
```

    IPython console for SymPy 1.5.1 (Python 3.6.9-64-bit) (ground types: gmpy)
    
    These commands were executed:
    >>> from __future__ import division
    >>> from sympy import *
    >>> x, y, z, t = symbols('x y z t')
    >>> k, m, n = symbols('k m n', integer=True)
    >>> f, g, h = symbols('f g h', cls=Function)
    >>> init_printing()
    
    Documentation can be found at https://docs.sympy.org/1.5.1/
    



```python
#init_printing?
```

### Parameters and two Gaussians


```python
rc, a, a1, a2 = symbols('rc, a, a1 a2', positive=True, real=True)
```


```python
u1=x**2*exp(-a1*x**2)
u2=x**2*exp(-a2*x**2)
gp=x**4*exp(-a*x**2)
wc=-1/x
wb=exp(-4*rc**2/x**2) - 1
u1, u2, gp, wc, wb
```

<br>

### Radial wavefunction $u$
We don't solve the radial equation for $R(r)$, but rather for $u(r) = r \cdot R(r)$.
So, if $R$ is a s-function, $u$ is a p-function, and if $R$ is a p-function, $u$ is a d-function, etc. <br>

$S = \int r^2dr R^2 = \int dr\, u^2$<br>
$T = \int r^2dr R_1 \hat{T} R_2 = \int dr\, u_1\, \frac{d^2}{dr^2} \, u_2 $
<br>

### Normalization constant


```python
N=integrate(u1*u1, (x, 0, oo))
N
```


```python
1/sqrt(N)
```


```python
printing.sstrrepr(1/sqrt(N))
```

### Coulomb


```python
W=integrate(gp*wc, (x, 0, oo))
W
```


```python
Ws=W.simplify()
Ws
```


```python
printing.sstrrepr(Ws)
```




    '-1/(2*a**2)'



### Soft-box


```python
W=integrate(gp*wb, (x, 0, oo))
W
```


```python
Wf=W.factor()
Wf
```


```python
Ws=W.simplify()
Ws
```


```python
Ws.expand()
```


```python
printing.sstrrepr(Ws.expand())
```




    '-3*sqrt(pi)*rc*sinh(4*sqrt(a)*rc)/(2*a**2) + 3*sqrt(pi)*rc*cosh(4*sqrt(a)*rc)/(2*a**2) - 2*sqrt(pi)*rc**2*sinh(4*sqrt(a)*rc)/a**(3/2) + 2*sqrt(pi)*rc**2*cosh(4*sqrt(a)*rc)/a**(3/2) - 3*sqrt(pi)*sinh(4*sqrt(a)*rc)/(8*a**(5/2)) + 3*sqrt(pi)*cosh(4*sqrt(a)*rc)/(8*a**(5/2)) - 3*sqrt(pi)/(8*a**(5/2))'




```python
(cosh(a)-sinh(a)).simplify()
```


```python

```

c1343d249b57c70f3215b7968f685f33b56d7a57
ipynb
Jupyter Notebook
notebooks/GTO_integrals/.ipynb_checkpoints/RAC_3D_P-checkpoint.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
notebooks/GTO_integrals/.ipynb_checkpoints/RAC_3D_P-checkpoint.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
notebooks/GTO_integrals/.ipynb_checkpoints/RAC_3D_P-checkpoint.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
Qwen/Qwen-72B
1. YES
2. YES