Verifying the shift/switch functions described in the GROMACS manual

Verifying the shift function described in the GROMACS manual.ipynb

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     "cell_type": "code",
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     "input": [
      "%matplotlib inline\n",
      "from __future__ import division\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "import sympy as sym"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
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     "cell_type": "markdown",
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      "# Nonbonded forces\n",
      "\n",
      "(Note: The [Gromacs Manual](http://www.gromacs.org/Documentation/Manual) gives an excellent description of all of this background material)\n",
      "\n",
      "The GROMACS force field is a typical molecular dynamics force field, including bonds, angles, torsions and non-bonded interactions. There are two types of non-bonded interactions: electrostatic and van der Waals. Electrostatics are modeled with a Coulomb potential:\n",
      "\n",
      "$V_c(r_{ij}) = f\\frac{q_iq_j}{\\epsilon_rr_{ij}}$ with $f = \\frac{1}{4\\pi\\epsilon_0} = 138.935485$\n",
      "\n",
      "and resulting force\n",
      "\n",
      "$\\vec{F}_j(\\vec{r}_{ij}) = f\\frac{q_iq_j}{\\epsilon_rr^2_{ij}}\\frac{\\vec{r}_{ij}}{r_{ij}} $\n",
      "\n",
      "van der Waals forces are modeled with a 6-12 Leonard-Jones potential\n",
      "\n",
      "$ V_{LJ}(r_{ij}) = \\frac{C^{(12)}_{ij}} {r^{12}_{ij}} - \\frac{C^{(6)}_{ij}} {r^{6}_{ij}}$ where the superscripted numbers in parenthesis denote the name of the constant (i.e. \"C twelve\")\n",
      "\n",
      "and resulting force \n",
      "\n",
      "$ \\vec{F}_{i}(\\vec{r}_{ij}) = \\left(12\\frac{C^{(12)}_{ij}} {r^{13}_{ij}} - 6\\frac{C^{(6)}_{ij}} {r^{7}_{ij}}\\right)\\frac{\\vec{r}_{ij}}{r_{ij}}$\n",
      "\n",
      "# Shift/switch\n",
      "\n",
      "In order to avoid calculating nonbonded interactions between all $N^2$ possible atom pairs, we want some way to say that, beyond a certain distance, the interaction forces are close enough to zero that we'll just call them zero. The obvious first choice is to pick a cutoff distance and set the potentials to zero at (and after) that distance. The obvious first problem with this is that the derivative of the potential (i.e. the force) would be infinite at the cutoff distance. You get all sorts of nasty effects, including heating. So, instead, you want to smoothly transition the potential to zero. While you're at it, you also want to smoothly transition the force to zero. GROMACS and CHARMM differ in what exactly to call the transition functions, but I'll stick with GROMACS nomenclature here and call this a \"shift\" function. Section 4 of the GROMACS manual works this out. In the end, we get the following (equations 4.27-4.30):\n",
      "\n",
      "For a pure Coulomb or Leonard-Jones interaction $F(r) = F_\\alpha(r) = r^{-(\\alpha + 1)}$ we can write the shifted force and shifted potential as\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "F_s(r)  & = & F_\\alpha(r) \\quad&r < r_1 \\\\\n",
      "F_s(r)  & = & \\frac{\\alpha}{r^{\\alpha+1}} + A(r-r_1)^2 + B(r-r_1)^3\n",
      "\\quad&r_1 \\le r < r_c \\\\\n",
      "F_s(r)  & = & 0 \\quad&rc \\le r\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "and (only middle one is in the manual)\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "\\Phi_s(r)  & = & \\Phi_\\alpha(r) - C \\quad&r < r_1 \\\\\n",
      "\\Phi_s(r)  & = & \\frac{1}{r^{\\alpha}} - \\frac{A}{3}(r-r_1)^3 + \\frac{B}{4}(r-r_1)^4 - C\n",
      "\\quad&r_1 \\le r < r_c \\\\\n",
      "\\Phi_s(r)  & = & 0 \\quad&rc \\le r\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "with\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "A & = & -\\frac{(\\alpha+4)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^2} \\\\\n",
      "B & = & \\frac{(\\alpha+3)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^3} \\\\\n",
      "C & = & \\frac{1}{r^{\\alpha}_c}  - \\frac{A}{3}(r_c-r_1)^3 - \\frac{B}{4}(r_c-r_1)^4\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "You can see that $C$ is chosen to zero out the potential at $r_c$.\n",
      "\n",
      "We have gently shifted the force to zero between a start value $r_1$ and a cutoff value $r_c$.\n",
      "This whole thing is also known as a \"force shift\" since we've applied a shift function\n",
      "\n",
      "$\n",
      "S(r) = A(r-r_1)^2 + B(r-r_1)^3\n",
      "$\n",
      "\n",
      "to the force in between $r_1$ and $r_c$\n",
      "\n",
      "The above formulas for $A$ and $B$ come from all of the versions of the GROMACS manual that I checked, as well as [JCTC2006]. I didn't find this until after doing the poking below, but the correct versions of the formulas are in [PCCP2010]\n",
      "\n",
      "[JCTC2006]:  David van der Spoel and Paul J. van Maaren, \"The Origin of Layer Structure Artifacts in Simulations of Liquid Water\", J. Chem Theory Comput. **2006**, 2, 1-11\n",
      "\n",
      "[PCCP2010]: Siewert J Marrink, Xavier Periole, D. Peter Tieleman and Alex H de Vries, \"Comment on 'On using a too large integration time step in molecular dynamics simulations of coarse-grained molecular models' by M. Wigner, D. Trzesniak, R. Barton and W. F. van Gunsteren, Phys. Chem. Chem. Phys., 2009, 11, 1934\", Phys. Chem. Chem. Phys, **2010**, 12, 2254-2256\n",
      "\n"
     ]
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     "source": [
      "# Checking on the electrostatics\n",
      "\n",
      "We can implement that quickly in Python and see what the results look like. Let's try to reproduce Figure 4.4 from the GROMACS manual, which shows the Coulomb force, Shifted Force, and Shift Function.\n",
      "\n",
      "We can compute the force two ways: we can use the explicit formula above, or we can take the numerical derivative of the potential. We'll do both."
     ]
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     "input": [
      "def get_abc(alpha,r1,rc):\n",
      "    \"\"\" NOTE: THIS IS WRONG, AS EXPLAINED LATER!\"\"\"\n",
      "    A = -((alpha+4)*rc-(alpha+1)*r1)/(rc**(alpha+2)*(rc-r1)**2)\n",
      "    B = ((alpha+3)*rc-(alpha+1)*r1)/(rc**(alpha+2)*(rc-r1)**3)\n",
      "    C = (1/rc**alpha) - (A/3)*(rc-r1)**3 - (B/4)*(rc-r1)**4\n",
      "    return A,B,C\n",
      "\n",
      "def get_s(r,alpha,r1,rc):\n",
      "    A,B,C = get_abc(alpha,r1,rc)\n",
      "    S = A*(r-r1)**2 + B*(r-r1)**3\n",
      "    return S\n",
      "\n",
      "def get_switched_force(r,alpha,r1,rc):\n",
      "    # Here, alpha should be 1 for electrostatics.\n",
      "    # The definition is F(r) = F_\\alpha(r) = r**(-(alpha+1))\n",
      "    unswitched = alpha/r**(alpha+1)\n",
      "    switched = unswitched + get_s(r,alpha,r1,rc)\n",
      "    switched[r<r1] = unswitched[r<r1]\n",
      "    switched[rc<=r] = 0.0\n",
      "    return switched\n",
      "    \n",
      "def get_switched_potential(r,alpha,r1,rc):\n",
      "    unswitched = 1/r**alpha\n",
      "    A,B,C = get_abc(alpha,r1,rc)\n",
      "    result = (1/r**alpha) - (A/3)*(r-r1)**3 - (B/4)*(r-r1)**4 - C\n",
      "    result[r>rc] = 0.0\n",
      "    result[r<r1] = unswitched[r<r1] - C\n",
      "    return result"
     ],
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     "input": [
      "# And reproducing Fig 4.4\n",
      "rmin,rmax,rstep=0.8,4.2,0.01\n",
      "r = np.arange(rmin,rmax,rstep)\n",
      "r1,rc  = 2.0,4.0\n",
      "\n",
      "analytic_noswitch = 1/r**2\n",
      "analytic_switch   = get_switched_force(r=r,alpha=1.0,r1=r1,rc=rc)\n",
      "computed_switch   = -np.gradient(get_switched_potential(r=r,alpha=1.0,r1=r1,rc=rc),rstep)\n",
      "\n",
      "fig = plt.figure()\n",
      "ax = plt.subplot(1,1,1)\n",
      "plt.plot(r,analytic_noswitch,'k-',label='Normal Force')\n",
      "plt.plot(r,analytic_switch,'g-',linewidth=1,label='Shifted Force')\n",
      "plt.plot(r,computed_switch,'r--',linewidth=2,label='Shifted Force from Potential')\n",
      "plt.plot(r,analytic_switch-analytic_noswitch,'k--',label='Shift function')\n",
      "ax.set_ylabel('f(r)')\n",
      "ax.set_xlabel('r')\n",
      "leg = ax.legend(fancybox=True,numpoints=1,loc='best')\n",
      "leg.get_frame().set_alpha(0.5)\n",
      "plt.grid(True)\n",
      "\n",
      "ax.set_ylim(-0.5,1.5)\n",
      "ax.set_xlim(0,rmax+1)\n",
      "plt.title('Comparing Coulomb Force, $r_1={r1}$, $r_c={rc}$'.format(r1=r1,rc=rc))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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        "<matplotlib.text.Text at 0x7fc263b1a0d0>"
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JyaSmpjJs2DBredtZ/uzFf/HiRevUnraJ29OnT9O6dWuSk5OJjo62Tu35888/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\nQtp6KuUAAAAISURBVDjq/wFfNZ/K+rLrngAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc263ba6210>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "They picked their limits well; the shift function (especially the shift function you'd get when computing it directly from the potential) can do some crazy things at lower values of r. Those kinds of numerical errors are handled by the MD program, and are a topic for another time."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Checking on the VDW\n",
      "\n",
      "The electrostatics look good. What about the van der Waals?\n",
      "\n",
      "Here, I'll make two plots, one near the minimum, and one near $r_c$. This came up in the context of [MARTINI](http://md.chem.rug.nl/cgmartini/) for me, so I'll use a MARTINI-related example, two NC3 particles. Those are type Q0, which means the LJ parameters are $C^{(6)} = 0.15091$, $C^{(12)} = 0.16267\\times 10^{-2}$. I also ran this example through GROMACS, and stored the results in the file `gromacs-results-shift-0.15091-0.0016267.dat`, which I load up below for comparison. I've worked with CHARMM long enough that I tend to keep measurements in Angstroms instead of nanometers, so you'll see a couple of factors of 10 to convert things below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "canned_grom_s = np.array([[  1.00000000e+00,   1.62655066e+09,   7.97663190e+01],       [  1.10000000e+00,   5.18231616e+08,   7.13482740e+01],       [  1.20000000e+00,   1.82394544e+08,   6.43340610e+01],       [  1.30000000e+00,   6.97899280e+07,   5.83998490e+01],       [  1.40000000e+00,   2.86725980e+07,   5.33143350e+01],       [  1.50000000e+00,   1.25242920e+07,   4.89079400e+01],       [  1.60000000e+00,   5.77020450e+06,   4.50534320e+01],       [  1.70000000e+00,   2.78578025e+06,   4.16535640e+01],       [  1.80000000e+00,   1.40173312e+06,   3.86326940e+01],       [  1.90000000e+00,   7.31753312e+05,   3.59311030e+01],       [  2.00000000e+00,   3.94786031e+05,   3.35010300e+01],       [  2.10000000e+00,   2.19384953e+05,   3.13038060e+01],       [  2.20000000e+00,   1.25211258e+05,   2.93078100e+01],       [  2.30000000e+00,   7.32095391e+04,   2.74869120e+01],       [  2.40000000e+00,   4.37526797e+04,   2.58193510e+01],       [  2.50000000e+00,   2.66734902e+04,   2.42868460e+01],       [  2.60000000e+00,   1.65577715e+04,   2.28739190e+01],       [  2.70000000e+00,   1.04484170e+04,   2.15674110e+01],       [  2.80000000e+00,   6.69197510e+03,   2.03560180e+01],       [  2.90000000e+00,   4.34399707e+03,   1.92300300e+01],       [  3.00000000e+00,   2.85401245e+03,   1.81810050e+01],       [  3.10000000e+00,   1.89528650e+03,   1.72016090e+01],       [  3.20000000e+00,   1.27049036e+03,   1.62854080e+01],       [  3.30000000e+00,   8.58550842e+02,   1.54267680e+01],       [  3.40000000e+00,   5.84058655e+02,   1.46207150e+01],       [  3.50000000e+00,   3.99388000e+02,   1.38628320e+01],       [  3.60000000e+00,   2.74075012e+02,   1.31492150e+01],       [  3.70000000e+00,   1.88389023e+02,   1.24763520e+01],       [  3.80000000e+00,   1.29412094e+02,   1.18411350e+01],       [  3.90000000e+00,   8.85925290e+01,   1.12407570e+01],       [  4.00000000e+00,   6.02145960e+01,   1.06726840e+01],       [  4.10000000e+00,   4.04234580e+01,   1.01346480e+01],       [  4.20000000e+00,   2.65962160e+01,   9.62458600e+00],       [  4.30000000e+00,   1.69344880e+01,   9.14063300e+00],       [  4.40000000e+00,   1.01959190e+01,   8.68108300e+00],       [  4.50000000e+00,   5.51683200e+00,   8.24439600e+00],       [  4.60000000e+00,   2.29282800e+00,   7.82914600e+00],       [  4.70000000e+00,   9.89690000e-02,   7.43405300e+00],       [  4.80000000e+00,  -1.36525900e+00,   7.05792000e+00],       [  4.90000000e+00,  -2.31307700e+00,   6.69966200e+00],       [  5.00000000e+00,  -2.89638000e+00,   6.35827300e+00],       [  5.10000000e+00,  -3.22364500e+00,   6.03282300e+00],       [  5.20000000e+00,  -3.37248500e+00,   5.72246100e+00],       [  5.30000000e+00,  -3.39849400e+00,   5.42639400e+00],       [  5.40000000e+00,  -3.34148000e+00,   5.14388800e+00],       [  5.50000000e+00,  -3.22990500e+00,   4.87425900e+00],       [  5.60000000e+00,  -3.08405200e+00,   4.61687000e+00],       [  5.70000000e+00,  -2.91831100e+00,   4.37113900e+00],       [  5.80000000e+00,  -2.74279300e+00,   4.13650600e+00],       [  5.90000000e+00,  -2.56453300e+00,   3.91245800e+00],       [  6.00000000e+00,  -2.38833400e+00,   3.69851400e+00],       [  6.10000000e+00,  -2.21739400e+00,   3.49422000e+00],       [  6.20000000e+00,  -2.05375600e+00,   3.29915300e+00],       [  6.30000000e+00,  -1.89863500e+00,   3.11291100e+00],       [  6.40000000e+00,  -1.75266700e+00,   2.93512100e+00],       [  6.50000000e+00,  -1.61607600e+00,   2.76543000e+00],       [  6.60000000e+00,  -1.48879800e+00,   2.60350000e+00],       [  6.70000000e+00,  -1.37058900e+00,   2.44901900e+00],       [  6.80000000e+00,  -1.26107300e+00,   2.30168600e+00],       [  6.90000000e+00,  -1.15980400e+00,   2.16121700e+00],       [  7.00000000e+00,  -1.06629300e+00,   2.02734300e+00],       [  7.10000000e+00,  -9.80035000e-01,   1.89981000e+00],       [  7.20000000e+00,  -9.00527000e-01,   1.77837500e+00],       [  7.30000000e+00,  -8.27273000e-01,   1.66280100e+00],       [  7.40000000e+00,  -7.59801000e-01,   1.55287200e+00],       [  7.50000000e+00,  -6.97663000e-01,   1.44837600e+00],       [  7.60000000e+00,  -6.40432000e-01,   1.34910700e+00],       [  7.70000000e+00,  -5.87716000e-01,   1.25487400e+00],       [  7.80000000e+00,  -5.39147000e-01,   1.16549200e+00],       [  7.90000000e+00,  -4.94382000e-01,   1.08077900e+00],       [  8.00000000e+00,  -4.53109000e-01,   1.00056500e+00],       [  8.10000000e+00,  -4.15040000e-01,   9.24684000e-01],       [  8.20000000e+00,  -3.79907000e-01,   8.52977000e-01],       [  8.30000000e+00,  -3.47469000e-01,   7.85290000e-01],       [  8.40000000e+00,  -3.17502000e-01,   7.21472000e-01],       [  8.50000000e+00,  -2.89802000e-01,   6.61381000e-01],       [  8.60000000e+00,  -2.64182000e-01,   6.04877000e-01],       [  8.70000000e+00,  -2.40472000e-01,   5.51823000e-01],       [  8.80000000e+00,  -2.18516000e-01,   5.02087000e-01],       [  8.90000000e+00,  -1.98172000e-01,   4.55542000e-01],       [  9.00000000e+00,  -1.79309000e-01,   4.12063000e-01],       [  9.10000000e+00,  -1.61813000e-01,   3.71527000e-01],       [  9.20000000e+00,  -1.45595000e-01,   3.33814000e-01],       [  9.30000000e+00,  -1.30580000e-01,   2.98810000e-01],       [  9.40000000e+00,  -1.16698000e-01,   2.66400000e-01],       [  9.50000000e+00,  -1.03885000e-01,   2.36473000e-01],       [  9.60000000e+00,  -9.20810000e-02,   2.08918000e-01],       [  9.70000000e+00,  -8.12330000e-02,   1.83629000e-01],       [  9.80000000e+00,  -7.12880000e-02,   1.60499000e-01],       [  9.90000000e+00,  -6.22000000e-02,   1.39426000e-01],       [  1.00000000e+01,  -5.39230000e-02,   1.20306000e-01],       [  1.01000000e+01,  -4.64140000e-02,   1.03039000e-01],       [  1.02000000e+01,  -3.96340000e-02,   8.75240000e-02],       [  1.03000000e+01,  -3.35420000e-02,   7.36650000e-02],       [  1.04000000e+01,  -2.81010000e-02,   6.13620000e-02],       [  1.05000000e+01,  -2.32750000e-02,   5.05210000e-02],       [  1.06000000e+01,  -1.90260000e-02,   4.10450000e-02],       [  1.07000000e+01,  -1.53200000e-02,   3.28410000e-02],       [  1.08000000e+01,  -1.21200000e-02,   2.58150000e-02],       [  1.09000000e+01,  -9.39300000e-03,   1.98730000e-02],       [  1.10000000e+01,  -7.10100000e-03,   1.49230000e-02],       [  1.11000000e+01,  -5.21000000e-03,   1.08740000e-02],       [  1.12000000e+01,  -3.68400000e-03,   7.63400000e-03],       [  1.13000000e+01,  -2.48500000e-03,   5.11300000e-03],       [  1.14000000e+01,  -1.57500000e-03,   3.21900000e-03],       [  1.15000000e+01,  -9.18000000e-04,   1.86200000e-03],       [  1.16000000e+01,  -4.73000000e-04,   9.53000000e-04],       [  1.17000000e+01,  -2.01000000e-04,   4.02000000e-04],       [  1.18000000e+01,  -6.00000000e-05,   1.19000000e-04],       [  1.19000000e+01,  -8.00000000e-06,   1.50000000e-05],       [  1.20000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.21000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.22000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.23000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.24000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.25000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.26000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.27000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.28000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.29000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.30000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.31000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.32000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.33000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.34000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.35000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.36000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.37000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.38000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.39000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.40000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.41000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.42000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.43000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.44000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.45000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.46000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.47000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.48000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.49000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.50000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.51000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.52000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.53000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.54000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.55000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.56000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.57000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.58000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.59000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.60000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.61000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.62000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.63000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.64000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.65000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.66000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.67000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.68000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.69000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.70000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.71000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.72000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.73000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.74000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.75000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.76000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.77000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.78000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.79000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.80000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.81000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.82000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.83000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.84000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.85000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.86000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.87000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.88000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.89000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.90000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.91000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.92000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.93000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.94000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.95000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.96000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.97000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.98000000e+01,   0.00000000e+00,   0.00000000e+00],       [  1.99000000e+01,   0.00000000e+00,   0.00000000e+00]])\n",
      "print canned_grom_s.shape                          "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(190, 3)\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def v_vdw(r,c6,c12):\n",
      "    v_vdw = c12/r**12 - c6/r**6\n",
      "    return v_vdw\n",
      "\n",
      "def v_vdw_switch(r,r1,rc,c12,c6):\n",
      "    vs12 = get_switched_potential(r,alpha=12,r1=r1,rc=rc)\n",
      "    vs6 = get_switched_potential(r,alpha=6,r1=r1,rc=rc)\n",
      "    v_vdw_switch = c12*vs12 - c6*vs6\n",
      "    return v_vdw_switch\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_vdw_potentials():\n",
      "    c6 = 0.15091E-00\n",
      "    c12 = 0.16267E-02\n",
      "    try:\n",
      "        grom_s = np.loadtxt('gromacs-comparisons/dat/gromacs-results-shift-0.15091-0.0016267.dat')\n",
      "    except IOError:\n",
      "        grom_s = canned_grom_s\n",
      "    r = np.arange(1,20,0.1)\n",
      "    analytic_noswitch_potential = v_vdw(r=r*.1,c12=c12,c6=c6)\n",
      "    analytic_switch_potential = v_vdw_switch(r=r*.1,c12=c12,c6=c6,r1=0.9,rc=1.2)\n",
      "\n",
      "    fig = plt.figure()\n",
      "    ax = plt.subplot(111)\n",
      "    plt.plot(r,analytic_noswitch_potential,'k.-',label='Normal potential')\n",
      "    plt.plot(r,analytic_switch_potential,'g--',linewidth=3,label='Our shifted potential')\n",
      "    plt.plot(grom_s[:,0],grom_s[:,1],'r-',label='GROMACS shifted potential')\n",
      "    leg = ax.legend(fancybox=True)\n",
      "    leg.get_frame().set_alpha(0.5)\n",
      "    ax.set_ylim(-4,.1)\n",
      "    ax.set_ylim(-3.5,-3)\n",
      "    ax.set_xlim(5,5.5)\n",
      "    ax.set_ylabel('$V_{LJ}$')\n",
      "    ax.set_xlabel('r (Angstroms)')\n",
      "    plt.grid(True)\n",
      "\n",
      "    fig = plt.figure()\n",
      "    ax = plt.subplot(111)\n",
      "    plt.plot(r,analytic_noswitch_potential,'k-',label='Normal potential')\n",
      "    plt.plot(r,analytic_switch_potential,'g--',linewidth=3,label='Our shifted potential')\n",
      "    plt.plot(grom_s[:,0],grom_s[:,1],'r-',label='GROMACS shifted potential')\n",
      "    leg = ax.legend(fancybox=True)\n",
      "    leg.get_frame().set_alpha(0.5)\n",
      "    ax.set_ylim(-.1,.1)\n",
      "    ax.set_xlim(10,13)\n",
      "    ax.set_ylabel('$V_{LJ}$')\n",
      "    ax.set_xlabel('r (Angstroms)')\n",
      "    plt.grid(True)\n",
      "\n",
      "plot_vdw_potentials()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w9+5d6dq1qzRs2NB8vFatWjJy5Mgk5509e1YMBoOcOnVKRLQLUmhoqPj4+EhU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\nBEEQBEEQBEEQBEEQBMHm/D/0nvSmmB9s8AAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc263a5d7d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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g2/QeX17JnY0QQoiHyo87G2mzEUIIYXaSbHRE7++hSHy2TeKzb5JshBBCmJ20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\nCCGEEEIIC/l/A2o3Sp19D2UAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc263941650>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There are a few things to notice here\n",
      "\n",
      " * The bottom of the potential well is shifted. That's to be expected. When you shift the force between $r_1$ and $r_c$ you endup shifting the potential over the entire range. MARTINI has been parameterized to take this into account.\n",
      " * Our shifted potential does go to zero at $r_c$\n",
      " * By inspection, it sure doesn't look like the derivative of our shifted potential is zero on the left size of $r=12.0$.\n",
      " \n",
      "That last part is pretty worrying. Let's see if it's real, or just a visual artifact."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_vdw_forces():\n",
      "    c6 = 0.15091E-00\n",
      "    c12 = 0.16267E-02\n",
      "    try:\n",
      "        grom_s = np.loadtxt('gromacs-comparisons/dat/gromacs-results-shift-0.15091-0.0016267.dat')\n",
      "    except IOError:\n",
      "        grom_s = canned_grom_s\n",
      "    r = np.arange(1,20,0.1)\n",
      "    analytic_switch_potential = v_vdw_switch(r=r*.1,c12=c12,c6=c6,r1=0.9,rc=1.2)\n",
      "    computed_switched_force = -np.gradient(analytic_switch_potential,0.1)\n",
      "    gromacs_computed_switched_force = -np.gradient(grom_s[:,1],grom_s[:,0][1] - grom_s[:,0][0])\n",
      "\n",
      "    fig = plt.figure()\n",
      "    ax = plt.subplot(111)\n",
      "    plt.plot(r,computed_switched_force,'g--',linewidth=3,label='Shifted force from our potential')\n",
      "    plt.plot(grom_s[:,0],gromacs_computed_switched_force,'k-',label='Shifted force from GROMACS potential')\n",
      "    leg = ax.legend(fancybox=True,loc=4)\n",
      "    leg.get_frame().set_alpha(0.5)\n",
      "    ax.set_ylim(-.1,.001)\n",
      "    ax.set_xlim(10,13)\n",
      "    ax.set_ylabel('$F_{LJ}$')\n",
      "    ax.set_xlabel('r (Angstroms)')\n",
      "    plt.grid(True)\n",
      "\n",
      "plot_vdw_forces()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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eKPuGYbpdpXYc7nc424qN1tvIJT/bpvX8TCEjfTiJFS0CWA7MAsLQN7EJM5o2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\nE1j2ktBSbQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc2639b4750>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Eek. So, as we guessed, our forces aren't matching up properly (that diagonal line straddling $r=12.0$ would become vertical if we made the $r$ step size increasingly small; right now, it's set to 0.1). Meanwhile, GROMACS (the code, not the manual) is doing just fine.\n",
      "\n",
      "So, something is wrong with the formula. Let's take a look at the shifted force in the shifted region:\n",
      "\n",
      "$\n",
      "F_s(r)  = \\frac{\\alpha}{r^{\\alpha+1}} + A(r-r_1)^2 + B(r-r_1)^3\n",
      "$\n",
      "\n",
      "with\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "A & = & -\\frac{(\\alpha+4)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^2} \\\\\n",
      "B & = & \\frac{(\\alpha+3)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^3} \\\\\n",
      "C & = & \\frac{1}{r^{\\alpha}_c}  - \\frac{A}{3}(r_c-r_1)^3 + \\frac{B}{4}(r_c-r_1)^4\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "This should be zero at $r=r_c$. However, if we plug $r=r_c$ into $F_s(r)$ and simplify a bit, we get\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "F_s(r_c)  &=& \\frac{\\alpha}{r_c^{\\alpha+1}}& + A(r_c-r_1)^2 + B(r_c-r_1)^3 \\\\\n",
      "  &=& \\frac{\\alpha}{r_c^{\\alpha+1}}&  -\\frac{(\\alpha+4)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^2}(r_c-r_1)^2 + \\frac{(\\alpha+3)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^3}(r_c-r_1)^3 \\\\\n",
      "  &=& \\frac{\\alpha}{r_c^{\\alpha+1}}&  -\\frac{(\\alpha+4)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}} + \\frac{(\\alpha+3)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}} \\\\\n",
      "  &=& \\frac{\\alpha}{r_c^{\\alpha+1}}&  -\\frac{(\\alpha+4 - \\alpha - 3)r_c - (\\alpha+1 - \\alpha - 1)r_1}{r_c^{\\alpha + 2}} \\\\\n",
      "F_s(r_c)  &=& \\frac{\\alpha}{r_c^{\\alpha+1}}& - \\frac{1}{r_c^{\\alpha+1}}\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "which explains everything above. If we have $\\alpha=1$ (i.e. the electrostatics case), this goes to zero and matches up with what we expect (and with the explicit Coulomb functions given in the manual and papers). If we *don't*, however, we get a discontinuity in the force. Now that we know what's going on, we see that things will cancel properly if we stick a leading $\\alpha$ onto the last two terms in $F_s$. I.e. we replace $A$ with $\\alpha A$ and $B$ with $\\alpha B$:\n",
      "\n",
      "$\n",
      "\\begin{align*}\n",
      "A & = & -\\alpha\\frac{(\\alpha+4)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^2} \\\\\n",
      "B & = & \\alpha\\frac{(\\alpha+3)r_c - (\\alpha+1)r_1}{r_c^{\\alpha + 2}(r_c - r_1)^3} \\\\\n",
      "C & = & \\frac{1}{r^{\\alpha}_c}  - \\frac{A}{3}(r_c-r_1)^3 - \\frac{B}{4}(r_c-r_1)^4\n",
      "\\end{align*}\n",
      "$\n",
      "\n",
      "Let's try it:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def get_abc(alpha,r1,rc):\n",
      "    \"\"\" This is now correct \"\"\"\n",
      "    A = -((alpha+4)*rc-(alpha+1)*r1)/(rc**(alpha+2)*(rc-r1)**2)\n",
      "    A = alpha*A\n",
      "    B = ((alpha+3)*rc-(alpha+1)*r1)/(rc**(alpha+2)*(rc-r1)**3)\n",
      "    B = alpha*B\n",
      "    C = (1/rc**alpha) - (A/3)*(rc-r1)**3 - (B/4)*(rc-r1)**4\n",
      "    return A,B,C"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_vdw_potentials()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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cuTNWVlY0b96cVatWac+VvOzUjB8/nnHjxrFmzRpq1qxJ8eLFta9/XU7YBjit\nPhK5DXD6yUSSSY6OjnTt2pX58+frXVZ+0/yMqOODfz3wDYWZoTNRC2U66yRACBCCiKtXtbcz9ZNB\nhQsXfmOr3dDQUKKioihcuLD28VejAaOiooiIiKBAgQIpvpgdHBy0W9Amd/v2be3zr6hUKnr37s3K\nlSv55Zdf6N279xtfRr/88gvlypWjXLlyAHTu3Jm1a9eSlJSk3aL2VdmpMTExYciQIRw6dIjo6Gi+\n+uor+vXrx4ULF944NjPbAGdGbt0G+F2MZRtgmUj0MHr0aBYtWpTpD39yA6sPZHc1K0pFgdWZfwm6\nqP9cFZDzSJLL7vkCr7ba3bZt2zuPffWL7+rqyty5c5kyZYr2C7dp06Zs2bLljS+HwMBASpQoQdmy\nZVM8/sEHH3Dnzh3u3btH/fr13zjXqlWriIyMxMXFBRcXF3x8fHjw4AE7d+7kvffew9XVlU2bNqXr\nPRYoUIAhQ4ZgZ2fH+fPnUz3G2LcBTutzIbcBTj+ZSPRQpkwZGjVqxPLly/Uuy8bchv61PueHOjDt\nuA0FzQoqEKFkSLa2tkyaNIkhQ4awefNmYmJiUKvVnDp1KsVfnK//0jdt2pQyZcqwYMECAEaOHEl0\ndDT9+/fn7t27xMXFsW7dOqZPn57ml/KOHTvYvn37G48fPnyYK1eucPToUU6fPs3p06cJDw+nR48e\n2qug2bNnM2XKFAICAnjy5AlqtZpDhw4xaNAgAObMmcOBAwd4/vw5iYmJrFy5kqdPn1K1atU3zpeZ\nbYDTI69sA+zs7JwjtgGWiURPvr6+zJ49m8RE/fcVGVlnJG39ttEs0oxmKLO8tuwj0THEPJJXnw8/\nPz+KFClCkSJF+Pzzz/Hz86Nu3bpA6mP9fX19mTt3LgkJCdjb23Po0CHi4uLw8PDAwcGBOXPmsHr1\najp37qx9TfIyPDw8qFChwhvPrVq1ivbt21OoUCGcnJxwcnLC2dmZESNGsHPnTh4/fkzHjh3ZsGED\ny5cvp1ixYhQpUoSJEyfSvn17AAoVKsTo0aNxcXHB0dGRhQsXsnnz5lT/ss8J2wCn9bkwhm2AmzRp\nIrcBNqA0R2NkhYYNG4q1a9cqV+CECUIMGaJIUfv371ekHGOW3lFbV69ezdI4chJZFzqp1UVu3wZY\njtoyQq+WTRFKXUIOHw7r1sGDtDaJTD/ZR6Ij11TSkXWhI+tCfzKRKKBVq1bExcWxb98+ZQosUgQ6\ndoSf5J5eul6NAAAgAElEQVTukmQIsjkoY2QiUYCJiQm+vr74+fkpV+iYMfDTT8Q+vs+TF08yXYzs\nI9GR+07oyLrQSa0u5DbAGSMTiUJ69OhBeHg4p0+fVqS8e8XtOF/Ojsl93fH7S8EEJUmSpDCZSBRS\noEABRowYgb+/vyLlHYw8yIDyF/n8wDMWHfmJp/GZm6uSF/pIVCpVihVv0yLbwnVkXejktbpISkpS\nvOlOJhIFDRo0iF27dqWYyJRZn7z3CXcrl+aOJTQ++ZjlJ/Wfq5Jbubi4EBoamq5kIkl5WVJSEqGh\noW9dAiczcmuPklBsBFUGjRkzBrVazezZs/Uua8HRBfw+aygT/oQuY0pwacRlzEzMMlRGSEhIrr8q\nefLkCevXr+f27dtvHTl3586dFEtR5GWyLnTyUl2oVCpcXFzo1q0b1tbWqT5P7s0LGZZ9A7Jfc/36\ndWFvby8ePXqkd1mx8bHC8fvC4kJhRCNvxNozGZ+rkhfmkaSXrAsdWRc6si50yOQ8ktyaeV7WiWF4\ne3vz3nvvMX78eL3L+jbkW27MnsyAaw6IoB3UKf7u9XkkSZIyI7NXJDKRZIGwsDCaN2/OlStX3ljC\nO6MePnvIjXuXqFy3PezZA5UqKRSlJElSSplNJLKzPQt4enpSpUqVN7YwzYzCFoWp7F4bhg2DTIwI\nk/NIdGRd6Mi60JF1oT+ZSLKIr68v/v7+Kfai0MvgwbB9OyTbw0CSJMkYyKatrAuAmjVrMnHiRD7+\n+GNlCvXxgfz5QckZ9JIkSS/JPpKUDJ5IQLPx0Ny5c7VbquotMhKqVePemb9R2driWMhRmXIlSZKQ\nfSRGqUOHDty6dYvDhw8rUt4V6ySOeNrzQ/8KzDg0I12vke2/OrIudGRd6Mi60J9MJFnIzMyMUaNG\nKbK1KMCFBxcY9N4lhoYmseLIYqKeRylSriRJkj5k01YWi42NpWTJkhw6dIhy5crpVZYQgvcXvc9s\n/3DWeEL50dMZ/4H+c1UkSZJANm0ZrUKFCjF48GBmzZqld1kqlYoxdcfgVx/GhMLcIz8SlxinQJSS\nJEmZJxNJNhg2bBiBgYHcvXtX77K6e3bnfOWiJJhC1ZN3WX3m7XNVZPuvjqwLHVkXOrIu9CcTSTZw\ndHSke/fuzJs3T++y8pvmx6fuSH6ob8Ks087Uc62nQISSJEmZJ/tIssmlS5eoW7cuV69exdLSUq+y\nYl7E8DjmPq7VG8OmTVCzpkJRSpKUl8k+EiNXpkwZGjVqxPLl+u8rYlXACleHUjByJCg0IkySJCmz\nZCLJRr6+vsyePZvExERlChwwAPbtg8uX0zxEtv/qyLrQkXWhI+tCfzKRZKPatWvj5ubGxo0blSnQ\n0hIGDQIFNtGSJEnKLGPqI5kCfIxmY5WHQB/g+mvHuAKrAKeXxy0B5qZSltH1kbwSFBTExIkTOX78\nuDL7Jt+5Ax4eXArdSQGX4rjauOpfpiRJeVJu6CPxAyoDVYBtwKRUjkkARgIVgTrAUKBCdgWohFat\nWvHixQv27t2rSHlhqvv8XsWaNZ/XY+rBqYqUKUmSlBHGlEhikt22BB6kcswd4NTL20+B80DRLI5L\nUSYmJowZM0axZVMePX/E8IqRDD4KgUcDuPs05VwV2f6rI+tCR9aFjqwL/RlTIgGYBlwDvIHv33Gs\nO1AVOJLFMSmuR48ehIeHc/r0ab3LaujWEJv3axLqCt2PxzP/n/kKRChJkpR+2d1HsgcoksrjE4Ad\nye5/CZQH+qZRjiUQAkxF0wz2OqPtI3nFz8+PM2fOKLKL4qZzm5g9qzO/bIXavrZEjL6OZX795qpI\nkpT3ZLaPxEz5UN6qWTqPWwvsSuO5fMBmYDWpJxEA+vTpg7u7OwC2trZUqVIFLy8vQHcpa8j7Hh4e\n/O9//+PatWtcuXJFr/Ls7tgRaVGUO5a3aHzyMeOXjqejR0ejer/yvrwv7xvf/ZCQEAICAgC035eZ\nYUyjtsoC/728PRyoBfR67RgVsBLNqK6RbynL6K9IAMaMGYNarWa2AsN3FxxdwKG5Y/j+qA2mR45S\nzKY4oPnQvPoA5XWyLnRkXejIutDJDaO2ZgBhaDrTvYDRLx8vCux8ebs+8CnQGDj58qdFtkapIB8f\nHwICAoiK0n9fkX5V+7Fw8U1KCGuKnbykQHSSJEnpY0xXJErKEVckAN7e3rz33nuMH6/QviI//wzb\ntsHOne8+VpIkKRm5Z3tKOSaRhIWF0bx5c65cuYK5ubn+BcbFQcmSsGcPVKqkf3mSJOUZuaFpK0/y\n9PSkSpUqiozeAsDcHIYPB39/QI6RT07WhY6sCx1ZF/qTicQI+Pr64u/vj1qtVqbAwYMR27dz5O/N\nXIu+pkyZkiRJaZCJxAh4eXlhaWlJUFCQIuX9HfsvG6oV4ODoTvye9LsiZeYGcmSOjqwLHVkX+pOJ\nxAioVCrGjh2Ln5+fIuWZqkwZ53mHfidh17F18qpEkqQsJROJkejQoQO3bt3i8OHDepdVs1hNSlXx\nYncZaPa7mu8OfEdOGXyQlWRbuI6sCx1ZF/qTicRImJmZMWrUKMUWcxxbbyx+9aFbOBzftYwJeyfI\nZCJJUpaQw3+NSGxsLCVLluTQoUOUK1dOr7KEEHQM7Ihqy1Z+2gnr/vcpPkNWKbMHiiRJuZKcR5JS\njkwkAJMmTeLOnTssXrxY77JeJL6gy6YutD8n6LP4CKpdu6B6dQWilCQp14mJQWVtDXIeSc43bNgw\nAgMDuXv37rsPfocCZgUY7jicPt/+imrJEmjVCo4dUyDKnEm2hevIutDJy3URnxSvuXH6NNSokely\nZCIxMo6OjnTv3p158+YpUp6ZqZnmcrVdO83yKa1bw9GjipQtSVLOdeL2CSr+5MGpyZ9D06YwcWKm\ny5JNW0bo0qVL1K1bl6tXr2JpqfC+IkFB0L8/d9f+zNJ8YUz4YILsN5GkPEQIwU9Hf+Lb7aOYvy0B\nj0cm2G//g2I1Gmf7EimpbU4lKaRMmTI0atSIZcuWKV94mzY8mO+HWftP2LHqa74I/kKO5pKkPOJx\n3GM6bexEwNLh/L0ggYcFofHnBQm3jder3MwmkpVoln1vDVjrFYGUKl9fX3744QcSExP1Kie19t/v\nbcL4tJ2a7evg2Jb5DNs1LE8kk7zcFv46WRc6eakuPlnfnuIrtrBrDXzZFJYOqMrhoSdpXqa5XuVm\nNpGMQJNIVMB4oJ9eUUhvqF27Nm5ubmzcuFHxsr9v+j22n3Sj9yfw6zo4tWUBQ3YOQS0UWutLkiTj\n8+gRm9cJep2BOgOgSJ9hhPYPpWzhsnoXrVTjeDvgV4XKUkKO7iN5JSgoiG+++YYTJ04o3o+RqE6k\n99bePNi2jtVboENXGDvuVz4u/7Gi55EkyQj8/Td06waffMLCjm442bvS0aPjG4cZeh5JSyBYobKU\nkCsSiVqtxtPTkx9//JGmTZsqXn6iOhHvbd7c27aWrTssKPRrMKqGDRU/jyRJBqJWw6xZmm0llizR\njN58C0PuR1Id+FuBcqTXmJiYMGbMGL2WTXlb+6+ZiRmr2q9i4LhACm38FVWnTnDgQKbPZezyUlv4\nu8i60MltdaEWamb8OYOJG4dA27awZQv88887k4g+MptIkjeqHQcm6x+KlJoePXoQHh7OqVOnsqR8\nUxNTOlfsjKppU1i/Hjp3hlz2iyVJecW92Hu0XNOS4OUT+GzAQv4rag4HD4KbW5aeNz2XMK2AXa89\nthvYBBwCLgCNgf3KhqaXXNG09Yqfnx9nzpxRbhfFtwkJgS5dYP167tTywNHCEVMT06w/ryRJetl/\ndT+fbupO3+C7DPsH+rYHs9Zt2NF9R7rLyMo+krNAZ+Bcssc8AEugAVABcAVaZPTkWShXJZLo6GhK\nlSrFiRMncMvivywAOHCApE4d6dXZFLNmzVnRboVMJpJkxDae3cgXK7uyarOgQCJ07wTercbzrde3\n5DPNl+5ysrKP5FMgP9AT3UTEc8A/wGzgM2BsRk8spZ+NjQ19+/Zlzpw5GX5tZtp/79WoQPduZsxZ\neY9bW3/Be5s3SeqkDJdjbHJbW7g+ZF3o5Ia6aB5pxsklJvxdHLoOdmD54N1MbzI9Q0lEH+lJJCeB\nU8AaoCLQHSj02jFnFI5Leo2Pjw8rV64kKioqy8/lYOGA3Uft6NAV1m6Ge1vX0HtbbxLV+k2OlCRJ\nYUlJMGkS1gOG8mjhbP4c0IzjQ0/rPcEwo9JzCeMI3E923xTNvBE1sP3lv8YmVzVtveLt7c17773H\n+PHjs/xcaqFm6M6hnNm6iK3roVcHsG/fjTUd1mCikmt9SpLB3boFPXqAqSmsWQNF9F+5Kiv7SH4B\n9qLpByme7F974C+ga0ZPmg1yZSIJCwvjo48+4urVq5ibm2f5+YQQDNs1jJNbFrBtPeyd1IvuY1dl\n+XklSUrdjSc3mPHnDObQgnwDBsKQITBhgiaZKCAr+0jKA27ALWAL4At8gCaZGGMSybU8PT2pWrVq\nhkZv6dP+q1KpmN9qPtU6DmXHzAF0998Nu14fwJdz5Ia2cKXIutDJKXWx8+JOavxUGdcZC4jt00Mz\nXP+bbxRLIvowS8cxfUg5YksyIF9fXwYPHky/fv0wMcn6JiaVSsW8lvM0f6lUH6CZ4LR8ObRpk+Xn\nliRJs/nUV3u/YkOwP5s3w5MC8F6/Z/zlWZzShg7updy6EUWubNoCTXNTzZo1+eabb2iXhTNV0/TP\nP5pksnSp5l9JkrLM47jHtFjdAqe9R/h5B8yqC+uaF2Vt5/V84PaB4ucz9FpbxibXJhKAwMBA5s6d\ny6FDhwwTwLFj0Lo1t/wn853daea2nEt+0/yGiUWScjHx4gXbO3hQ5a8rdO8Idk1asbL9ShwsHLLk\nfIZca0vKZh06dODWrVuEhoa+89gsaf+tUYPr6xaTb8gw7q5eTJeNXXR7PxuxnNIWnh1kXegYbV1c\nvYrqgw9oqSrHJ+Pc6NB/Jju678iyJKIPmUhyIDMzM0aNGqXXYo76+iXfOZr3ULMwCEy2/kqnwE68\nSHxhsHgkKVfZtAlq14YePci/YxehvhcYU2+M0Q69l01bOVRsbCwlS5bk0KFDlCtXLtvPL4Tgyz++\n5PfNfgSvhmGtIK59azZ32UwBswLZHo8k5XSBZwOpZf8+7lPnQXAwbNgANWtmawyyaSuPKVSoEIMH\nD2bWrFkGOb9KpeL7pt/TotOXNO8F83dBwW072XEx/QvESZIEzxOeM2jHIL5a3JUXtaujvnsHTpzI\n9iSiD5lIcrBhw4YRGBjI3bt30zwmK9t/VSoV05tMp02nCXzUC1bss6JTmPGuyWW0beEGIOtCx5B1\nceHBBWovrc2TlUsIXQZz3n/GjOFVwdbWYDFlhkwkOZijoyPdu3dn3rx5BotBpVIx9cOpzP/yAJb7\n/wIfH1i3zmDxSFJOsfrMahrMr8bwpWF8ux+a9YJHfbowrPZwQ4eWYbKPJIe7dOkSdevW5erVq1ha\nWho6HAgPh48+0mzt2aOHoaORJKMVsG4c1X38OOMMI9rlZ1q7uQysPvBVP4VB5IZ5JFOAjwEBPEQz\no/76a8eYAweAAmiWtv8VSG0FwzyTSAA6d+5MgwYNGDFihKFD0Th7Fpo1Az8/Lreqi4uVCxb5LAwd\nlSQZj5UrEWPGMP+TYvzk+ZzALht53/l9Q0eV6URiTKyS3R4OLE3juFffSGZo9opvkMoxIi85cuSI\ncHNzEwkJCW88t3///uwPSAghzp4V8UWcxLCu1qLJyiYiNj7WMHEkY7C6MEKyLnSytS5iYoTo3VuI\nChWECAsTj549EjEvYrLv/O+A5g/5DDOmPpKYZLctgQdpHPfs5b/50Sxp/ygrg8oJatWqhZubGxs3\nbjR0KFqPShbhw95qvtz1hGJb99JmbRti42MNHZYkZbuYFzEcuXEEzpzRjMQyNYWjR6FSJewK2mGZ\n3wiapHOZacA1NPvApzVswQTNRlsxgF8axxg6sWe7oKAgUaVKFaFWqw0dita0g9NEuWGI61aIPu0Q\nXgFe4umLp4YOS5KyzcnbJ0W5uWXFiE8KisTC9kKsWmXokN6KHHJFsgcIS+Xn1ep/XwElgADghzTK\nUANV0Cxj3xDwyrJoc5CWLVsSHx/P3r17DR2K1oQPJtC36ww+9Ibv9kOpLSG0Xtta7rQo5XpCCBYe\nXUiz+bWZvOQ/+v31nB4jipPUM3cOQEnPMvJKapbO49YC79r4IhrYCdQAQl5/sk+fPri7uwNga2tL\nlSpV8PLyAnTjxnPb/TFjxjBz5kzMzMy0zycfI2+I+L5s8CVXT16lTpMl/L0XLnoU4dDBQwaJ5/U6\nMfT/lyHvnzp1Ch8fH6OJx5D358yZo+j3Q9DvQfiH+vPk9gFCN8FCB/i8lTkL24/D1MTU4O83+f2Q\nkBACAgIAtN+XOV3ZZLeHo9mZ8XUO6Jq8CgIHgSapHGfoK0SDiIuLE0WLFhUnT57UPmYsnaoz/5op\nlgdOEMLVVYjFiw0Sg7HUhTGQdaGjdF38c/2I+KK1ibhngejaEVF5YWXx74N/FT1HViGTTVvGNMxr\nE5rdGJOAy8Bg4B5QFPgZaA28j6bZy+Tlzy9AaisXvqyTvMfPz48zZ85kaBfFbHX5Mnz4IYwfD59/\nbuhoJElZUVHQvz93z/1D/Y9u0rz5EGY1n4W5WdZvja2E3DCPREl5NpFER0dTqlQpTpw4gZubm6HD\nSd2VK5pkMnasZs9pScoNjhyBbt2gbVuEnx8H7xyhkXsjQ0eVIXLRRgkAGxsb+vXrx5w5cwAjXVOp\nVCnYvx/8/Lg1fQJt17UlOi46y09rlHVhILIudPSpi8jHkSAEzJql2TF09myYOxeVuXmOSyL6kIkk\nFxoxYgQrV64kKirK0KGkrWRJ/t20mAT//1FyVRAfrf6Ix3GPDR2VJKWLWqjx+8uPWv8rzZ0Pa0Fg\noGYb6k8+MXRoBiGbtnIpb29vypcvz4QJEwwdSpqWnVjGd78MYH8AzK0NoR1r8nuv37E1z1krn0p5\ny/3Y+3hv8yZ6XzDrNsGvlQvw8aYw3JzKvvvFRi6zTVvZPfxXyiZjxozho48+YtSoUZibG2dHX/9q\n/YlPiseLIexfCSqO0oxm/P7p79gVtDN0eJL0hoORB+mxsRu9dt9mxN8w4GN42KQq7Qrk7c3cZNNW\nLuXp6UnVqlX55ptvDB3KWw2uOZivei7Cqw8M/QcabjrGH1f+yJJzyX4BHVkXOumti4SkBMas7s2y\nRbdpfRFqDoSKfcdysM9BStiUyNogjZy8IsnFTExMmDNnDseOHWPr1q3YGulmOYNqDMJEZUJjBhK+\nyRGbXRFQ0dBRSVJK+Q4e4tC8WGaVV/FjS3tWdPyFlmVbGjosoyD7SHIxLy8vDhw4AEDr1q0JCgoy\ncERvd+HBBd57YQWNG0P//jBunKFDkiRISoKpU2HxYggIILDYY+q71qeYdTFDR6Y42UcivcHCQrPi\nvqurK+Hh4URGRhrv3BLgPYf3NDf279fMM1GrNRMXJSmbJaoTUQs1+e89hJ49NQ8ePw4uLnQxbGhZ\nZsCAAZl+rewjycXWrl1Lo0aNOHPmDD4+PjRu3JiIiAhDh/VuxYppksnKlTBtGqfvnOZ+7H29i5X9\nAjqyLnRer4ubT27SZFUTln/fDapVg0aNYM8ecHExTIDZ4Pbt22zatCnTr5eJJBeztbVl8uTJ2Nra\n4uPjk7OSSdGisH8/cQFL2dm7Dh+u+pB7sfcMHZWUywX/F0z1BZX5aPlB2v5vK399PxQmTdLsIZJL\n7dmzh+rVq+Po6GjoUIyO4VY9M3Lz5s0Tbm5u4sqVK4YO5Z2i46JFxW8Ki7MOiIleiIo/VRR3n941\ndFhSLhSfGC/G/j5WFB+J+NMVsbs0wtlXJWaFzjJ0aFkmISFBfPXVV6Jo0aJi7969IioqKtOLNso+\nkjxm2LBhqFQqvLy82L9/P6VKlTJ0SGmyLmDNl53m0CSuN38ECEzEWRoLL/Z578fZ0tnQ4Um5yOSQ\nyYQH+HH0V5hTB1a3cGFDp3W5dpmTGzdu0KNHD8zNzTlx4gTOzvL3KTWGTvZGI60lshcsWCBKlCgh\nLl26lL0BZcKaM2uEs69KhDkivm2IqDjfQ7xIfJHhcuTS6TqyLnT2//67eDZiqLhhayrq90W0WN1C\n3Ht6z9BhZZmgoCDh7Owspk+fLpKSklI8h7wikTJi8ODBmJiY0LhxY/bt20eZMmUMHVKaenj2wLS3\nKU1VPfg9QE27U6XIb5LP0GFJOVSSOglTE1PNYouhoTBiBAVLl+bWgSDaPjmNb31fTFS5r/s4ISGB\nCRMmsGHDBjZt2kSDBg0UK1vOI8njfv75Z7777jv27dtH2bLGvVZQ4NlAEu7coueoFdC6NUybBqrc\n+hGWlBQdF8368PWsOLWCtoWq8VWEKwQEaD4/I0Zo9sbJxZ+liIgIunXrhoODAwEBATg4OKR6XGbn\nkeRWBrlkzKmWLl0qihUrJi5cuGDoUNLn/n0hKlcWYtw4IdRqQ0cjGakkdZL44/IfoufmnsJ6cgHR\nsTNiZxlEVEGVSBrQX4jQ0Dzx+dm6datwdHQU/v7+bzRlvY5MNm3lVtn0X2T80tsWvmzZMlGsWDFx\n/vz5rA1IKQ8eCFGlihC+vkKo1UKdji8E2S+gkxfq4sqjK6LKIMSPtRD3LBB73RGffoKw+dpMHL15\nVHtcbq2LuLg48cUXXwg3Nzdx+PDhdL0G2Uci6aNfv36YmJjQpEkT/vjjDypUqGDokN6ucGHYuxea\nNuX255/Sptp5tnX7FVcbV0NHJhnaw4ewZg0lV6wg+HoBFld6Qa3PwOa9yvSt0pcf3u+Jg0XqTTu5\nxeXLl+natSuurq6cPHkSOzu5mnZmZHGuz71WrlwpihYtKs6ePWvoUNLlWNjv4kQxEzGrDqLUnJIi\n8nGkoUOSsolarRaHrx8WA7cPFEcjDwuxc6cQnToJYWMjRI8eQuzZIzaFBYphO4eJE7dOGDrcbLNh\nwwbh6Ogo5s6dm64r9eSQVySSEnr37o2JiQlNmzZlz549VKxo3Mvw3s7/gn69Tdi5Uo3Juqt4iUbs\n7xOCm63xrikm6efO0zv8cvoXVpxaQeKF8/Q9BWXOrYYylaBvX/j5Z3i50nVHoGOlzoYNOJs8f/6c\nUaNGsWfPHoKDg6levbqhQ8rxsijX5zyZbf9dvXq1cHFxEWFhYcoGlAV2/LtDOE7IJ44U1bSHu//g\nJq5GXX3juNzaFp4ZObUutpzbImwmmIh+HyMOuSJuF0L41UPU8ikkYuNjM1VmTq2L5C5cuCDef/99\n0bVrVxEdHZ3pcsjkFUnuGywtKaJnz57MmjWLZs2aERYWZuhw3qpNuTas8N5K6775qHUTxqyL5MSt\n44YOS1KSWg0hIbSYup6I2WraXIT/1Yf3xllwYWw/Zo3cTUGzgoaO0iBWr15NgwYNGDp0KOvWrcPa\n2jrbY8it44VfJldJXxs2bMDHx4fdu3dTuXJlQ4fzVsH/BdPvl46E/VoMh/rNYP58MJF/K+U0j54/\nYvO5zfSr2g/TGzc1q0AHBEDBgtCvH30s/+Bq/lj6VulLJ49OWOa3NHTIBhEbG8vw4cMJDQ0lMDCQ\n999/X+8yMzuPRCYS6Z0CAwP54osv+O2334w+mdx9ehdndUFo0QIqV4affpLJJAdIUifxx5U/WH5q\nObvDttLqbALz7lTF4XwkdO2q6fuoUQNUKuKT4slvmt/QIRvU2bNn6dKlC9WrV2fBggVYWiqTTOWE\nxJQUannM+ZRq/924caNwdnYWJ0+eVKS8LPfkiRD16wsxcKAQLydh5Ya2cKUYU10sO7FMFJ9VTNT4\nDPFTDcSDgojfSiHm+NQR4vnzLD+/MdXFu6jVarFs2TLh4OAgli9fnuFRWe+CHLUlZaVOnTqhUqlo\n0aIFu3btolq1aoYO6e2srCA4GFq1gkGDCP3amxvRNwwdlfS6e/eouPo3gn+5ScFEWFEFqg4CZ48a\nDKreH8zNDR2h0YiJiWHw4MGcOnWKAwcO4OHhYeiQtHLrJczL5CopbevWrXz++efs2rUrZwwvfPqU\n6CYN+DXpLBO6OTKmwVgqOVXC08kTp0JOry7lpSwmhCDicQQl7UpCQoImyS9fDiEhJLRtTVvzLZws\nY0nPKr3oW6Uvns6ehg7ZqJw6dYquXbvSsGFDfvzxR+022kqTTVspKXq5J6W0bds24eTkJI4ePfru\ngw0sNj5WlJzqKPa7IYLKIoa1RDTyRhT2Rfx8/GdDh5frXY++LqYdnCbKzi0r6oy0EgkjRwjh7Kxp\ndly6VNMEKYQ4fut4prYGyO3UarVYsGCBcHBwEGvWrMny85HJpq3cmnle1okUEhKCl5eX4uVu376d\nzz77jKCgIGrWrKl4+UoKiQihc0Ar6u1+TstEqHQPPO9BQWt78leuBpUqgaen5l8PD4YcGMu16GtU\ncqqk/angUIECZgUM/VYUk1Wfi1e2nN/Czyd+5kj4b3QJE/Q9BcWfQHSXdnj4/g/Kl8+yc2dUVtdF\nZkVHR/PZZ59x8eJFAgMDKVeuXJafM7NXJLKPRMqUjz/+GBMTE9q0acP27dupXbu2oUNKk5e7FydH\nXeR7p+85XvwFAffCOHsvnBudD5D/v2sQHg4hITBvHvz7L19aJHLSIYEwp50EOcH3TnDJ0YSDAw9T\nq1gtQ78d46dWc+IXPz4NOsK6i7CnNHzbCP72sOK7Zk3xMKIkYqyOHTtG165dadGiBatWrcLcyPuK\n5BWJpJedO3fSt29ftm/fTp06dQwdTroJIVLtH3n2/AnVxtpQ6a7uyqXSPSgRDfnKV8DMs7Lu6sXT\nE9zc6LipM1b5rfB08tRewRS1Kpon+l9S1OPVq5r5HgEBPLYw5ZuSV1nrCVUrNqFvlb58UuETLPJl\nTdt+biGEYO7cuUybNo2ffvqJzp2zd3kXOY8kJZlIslFwcDDe3t78+uuv1K1b19Dh6CVJncSFBxcI\nu4jQvgUAABdeSURBVBdG+L1wwu+FE3YvDJPnL/iv2XbN1UtYmObf8HDE48f8Y/OUMCcId4IwZ82/\nCQ52RPpEYlXAytBvSXGJ6kR2X9rNilMrKKw2Z0l8c1ixQlMnPXpA377Ee3rgH+pPD88euNu6Gzrk\nHOHRo0f069ePmzdvsmHDBkqVKpXtMcjO9pSyvFMqp8iuMfLBwcHC0dFRHDp0KFvOlxn61EVaHcGn\nz4WI+n0Rg1oj5tdEhLghHpoj7hcyEaJxYyGGDxdiyRLNJkrR0eJ5wnPRcnVLMea3MSLgZIA4fuu4\neBb/LNNxZVZm6uL8/fNi7O9jRZGZzqJOf8SSaohH5oj4Fs2E2LhRiLg45QPNBsYwjyQ0NFS4ubkJ\nHx8fEWfAekTOI5EMqUWLFqxevZpPPvmELVu2KLoftDFIayZ16dI1mDpxP+H3wjlzN4y198MJvxtG\nK8uqrCv/pebqJTQUFi+G8+cxsbdhSMHbhDkH87sTzHKC/xxV1ChZnz/7/pnN7yr9YuNjaT2zKp1O\nxLH/pOZP1hVVoNIQmNPvMzpX7GToEHMktVqNv78/s2bNYsmSJbRr187QIWVKbr2EeZlcpey2Z88e\nevbsyebNm/nggw8MHY5BCCGIiY/BusBri+clJbEteA4rV4+h0su+F8+7UPIx3HMqhFuD1ilHkJUs\nycXHl5l6cKq278XTyZPi1sWzr/8lPh527IAVK4gN2cO6cvEsrwpXKzjTq3Jv+lbpSwVHI98EzUjd\nv38fb29vHj9+zLp163BzM/zWB7KPJCWZSAxo7969dO/enY0bN9KoUSNDh2NU7jy9w6Frhwi7G0b4\n/XDC7oZx7d5/THHpia9Ny5R9MPfv87CkM9vzXdX2wYQ7QayDFX2q9OXHlj8qGlvk40hWnl5J7WK1\naR5bRNPvsWYNVKwI/frxZw0nfghbQr+q/WhRpgVmJrJBI7MOHjxIz5496dGjB1OnTiVfvnyGDgnI\nHX0kU4DTwClgL/C2PVNNgZPAjjSeN1gbo7ExVPvv3r17haOjo1G0P79iTLEkFxsfKx4+e/jmE9HR\nYsGPvcSAtpp9Vva6a/Yef1AQ8V+lYkIMGSLEwoVC/PmnEFFRQggh9l3ZJ3yCfcSyE8vEkRtHxNMX\nT1M956u6eBb/TKw5s0Y0WdlE2I9FDG2J+M/dWghXVyG++UaIS5ey6m0bjez8XCQmJoopU6YIZ2dn\nsWvXrmw7b3qRC/pI/IBvXt4eDkwCBqRx7AjgHJD7hsTkEh9++CGBgYF06dKF9evX8+GHHxo6JKNl\nkc8i9WGx1tZ80G0s1K1L+L1wNt0LI+xeGPkfPGZF6W6UiSsOx45phtyePQu2thR1yY9LviuEOME8\nZ/jXAVycSvFl/S/5rPpnKYo/cfsETVc0pta5Jww8CR9dhl1lYWjDGFb8GE5R27f9LSdl1N27d/n0\n00958eIFx48fp1ixYoYOSTHGegkzHrABvkzlueJAADANGAW0TeWYl8lVMrQDBw7QuXNn1q1bR5Mm\nTQwdTo4nhOBWzC0K5S+Erbmt7gm1GiIjmfpTV56dPKqdA1PmEVyzgXyVq1Hqg7a6/hcgccUy7i30\n54alYEVV2FAJalVqTt8qfWn/XvtcNZPf0Pbu3Uvv3r3p168fkyZNwszMmP6G18ktfSTTgF7AM6AO\n8DiVYzYC0wFrYAwykRi9gwcP0qlTJ9auXUvTpk0NHU6utufyHg7fOKyd/xJx9yKlHqrZ6vH/9u49\nOooqT+D4FxFcjBBQI++nCc8gEQggHiAkamQY8MXLwEI6riwKapQ9iMywuuCOjiuCyo6KnFQSHlmQ\nVyAQIECigKjII6QxEliMAeEIDsIIkQ1J9/5xK+kO5EnSXV3Vv885nFRXV6p//aO6f6l7q+79D7qe\nLXL1wRQWQkwMf7n3LFrJfmxhNib3mUy7Zu2MfguWUlxczLx581i6dCnJyck+f/ybZYiUDKBVBevn\noPo7/qT/mw0sBGzXbfdH4ByqfySiqheKjY2lU6dOADRv3pywsLCy8XSysrIA/OJx6bKR8TgcDubO\nnUtMTAzLly+ncePGhsRTus6X/n/q+/HD9z5Mo1ONGBo0lIixEVwtvsryjcvJD+xA1+mPlG1/+PBh\n4uPjeanoCg/s/YYGJQ3KiogvvR9vPF60aJFHvh9CQkKIiYnh8uXLLF68uKyIGP1+r/9+SExMBCj7\nvrSSDoC9gvV/AU4BPwBngStAcgXbGdph5Ut8qYN5z549zqCgIOfWrVsNeX1fyoXRJBcunshFenq6\ns2XLls758+c7i4uL633/noIFRv8NAY7ryy8AA1DNXJUZhjRtmc6XX37J448/TlJSEiNGjDA6HCHq\n1bVr15g7dy7Lly9nxYoVprv8/WabtnxpMuu3gBzU5b8RwEx9fRtgcyW/I9XCZAYPHszGjRuZMmUK\nW7ZsMTocIepNQUEBERERZGdnc+jQIdMVkbrwpUIyBugNhAFPofpCAM4AIyvY/nNgtHdCMy/3/gFf\nMWjQIDZt2kRsbCxpaWlee11fzIVRJBcu9ZGLjRs3Eh4ezmOPPcbmzZsJCgqqe2Am4pvXoAnLGzhw\nIGlpaYwaNYqlS5cyalRFLZRC+LaioiJmz57N2rVrWb9+PYMHDzY6JEP4Uh9JfZI+EpP49ttvGTly\npKkHrBP+6eTJk0yYMIHWrVujaRp33nmn0SHVmRX6SIQf6t+/P1u2bGHq1KmsX7/e6HCEqJE1a9Yw\naNAgYmJi2LBhgyWKSF1IIbE4M7SF9+vXj/T0dJ577jnWrVvnsdcxQy68RXLhUptcXL16lenTpzNr\n1iw2b95MfHy8X8yEWR3pIxE+oW/fvqSnpzNixAgcDgdjxsj8FsK3HD9+nHHjxhEcHMyhQ4cIDAw0\nOiSfYdVSKn0kJpWdnU10dDQffvih1+erFqIyKSkpvPjii8ybN49p06ZZ9izELEOkCFGlPn36sH37\ndqKjo3E4HIwfP97okIQfKyws5KWXXuLzzz8nIyODsLAwo0PySdJHYnFmbAu/77772L59O/Hx8aSk\npNTbfs2YC0+RXLhUlovc3FwGDhxIYWEhBw4ckCJSBSkkwif17t2bjIwMZs6cycqVK40OR/iZpKQk\nhg4dSnx8PMuXL6dpU5n6qCrWbOiTPhLLOHr0KA8//DDvvPMOkyZNMjocYXGXL19m+vTp7N+/n9Wr\nVxOqz93iL+Q+EmFJvXr1YseOHbz66qskJ1c00LMQ9ePIkSOEh4fTsGFD9u/f73dFpC6kkFicFdrC\ne/bsyc6dO3nttddISkq66f1YIRf1RXLhkpmZyZIlS4iKimLOnDkkJCQQEBBgdFimIldtCVPo3r07\nu3btIioqCofDgc12/ZxnQtTepEmTSEtLo6SkhJ07dzJgwACjQzIl6SMRppKXl0dUVBRvvPEGzzzz\njNHhCBMqKipi06ZNaJrG1q1bKSkpAWDs2LGsXr3a4OiMJfeRCL/QtWtXdu3aRWRkJA6Hg2effdbo\nkIRJZGdno2kaK1eupGfPnthsNoqKisjIyKB///4sWbLE6BBNS/pILM6KbeEhISFkZmYyf/78Wn34\nrZiLm+Uvubhw4QKLFy+mX79+jBo1iqZNm7Jv3z6ysrKYMmUKq1evZtiwYWRkZNC8eXOjwzUtOSMR\nphQcHExmZmbZmcm0adOMDkn4iJKSEjIyMtA0jW3btjFixAjefvttIiMjadiwYbltmzdvzhtvvCFF\npI6kj0SY2smTJ4mMjGTWrFk8//zzRocjDHT8+HESExNJSkqidevWxMXFMWHCBFq0aGF0aKYhfSTC\nL3Xp0oXMzEyGDx+Ow+FgxowZRockvOi3337js88+Q9M08vLymDRpElu3bpV7QLxM+kgszh/awjt3\n7kxWVhYLFizggw8+qHQ7f8hFTZk5F06nky+++AKbzUaHDh1ITU1l5syZnD59mgULFtS6iJg5F75C\nzkiEJXTq1ImsrKyyM5P4+HijQxL17NSpUyQnJ5OYmEjjxo2x2Wy89dZbtGrVyujQ/J70kQhLKSgo\nYPjw4cyYMYOXX37Z6HBEHV29epXU1FQSEhLYv38/48aNIy4ujvDwcMvOCWIk6SMRAujQoUPZmYnT\n6eSVV14xOiRRS06nk4MHD5KQkMCqVasICwsjLi6ODRs20KRJE6PDExWQPhKL88f23/bt25OVlcVH\nH33Eu+++W7beH3NRGV/Mxfnz51m4cCF9+vRh7NixtGzZkgMHDrBjxw5iYmI8VkR8MRdmI2ckwpLa\ntWtXrs9k1qxZRockKlBcXEx6ejqaprFr1y5Gjx7N+++/z7Bhw7jlFvk71yys2sgofSQCgJ9++onI\nyEhsNhuzZ882Ohyhy83NRdM0li1bRufOnbHZbIwfP55mzZoZHZpfu9k+EikkwvLOnDlDz549CQgI\noE2bNiQnJ9O9e3fprPWyS5cusWrVKhISEigoKGDy5MnExsbSvXt3o0MTOikk5Ukh0WVlZREREWF0\nGIZ74IEH+OqrrwBo3LgxAQEBhIaG0rt377KfvXr18pu7oL11XDgcDjIzM9E0jbS0NB566CFsNhvR\n0dHceqtvtKzLZ8RFrtoSogqlBaJ///5kZGRQVFSE3W4nJyeHgwcPkpycjN1uJzAwsKy4lBaYHj16\nyNVCtZSfn09iYiKJiYkEBgYSFxfHokWLuPvuu40OTXiAnJEIv3Dx4kWmTp3KkiVLKh2gz+FwUFBQ\nQE5ODna7vazQHD9+nPbt25c7ewkNDSU4ONhn/qr2BYWFhaxbtw5N08jOzubpp5/GZrNx//33SzOi\nSUjTVnlSSES9uXbtGnl5eeWKi91u58yZM3Tr1q1ccQkNDaV9+/Z+88XpdDr5+uuvSUhIYM2aNQwc\nOJC4uDhGjx7NbbfdZnR4opakkJQnhUQn7b8u9Z2LK1eu8N1335UrLjk5ORQWFpZrGiv9edddd9Xb\na9dVXXNx9uxZli1bhqZpFBcXExcXx+TJk2nbtm39Bekl8hlxkT4SIbwsICCA8PBwwsPDy63/5Zdf\nOHr0aFlxSUlJwW63c/vtt99QXEqvJjODoqIi0tLS0DSNPXv28OSTT/Lpp5/y4IMP+s0ZmKiYVf/3\n5YxE+BSn08np06fLnbnY7XaOHTtG69atb+jg79q1K40aNTI6bACOHDmCpmmsWLGCHj16YLPZGDNm\nDHfccYfRoYl6Jk1b5UkhEaZQXFzMiRMnyhUXu91OQUEBISEhNzSRdezY0St3fF+4cIGUlBQ0TePn\nn39mypQpxMbGEhwc7PHXFsaxQiGZD4wGnMDfgVjgVAXb5QP/AEqAa8CACraRQqKT9l8XM+Xi999/\nJzc394YryC5dukSvXr1u6OBv2bJlrfZfUS5KSkrYsWMHCQkJbNu2jUcffZS4uDiioqJumKLWSsx0\nXHiaFfpI3gHm6ssvAK8D/1LBdk4gArjgnbDM7fDhw/Ih0ZkpF02aNKFv37707du33Ppff/21rLDY\n7XbWrl1LTk4OjRo1uqG4hIaG0rRp0wr3756LEydOoGkaycnJtGrVCpvNxscff+w3N2ea6bjwVb5U\nSH5zW74D+KWKbX3pTMqnXbx40egQfIYVctGiRQuGDBnCkCFDytY5nU7OnDlTVlz27t3LJ598Qm5u\nLkFBQTd08Hfr1o1z586haRqapnHs2DEmTpzIli1b6N27t4HvzhhWOC6M5kuFBOA/gX8GCoFBlWzj\nBHagmrY+AT71TmhC+KYGDRrQtm1b2rZtS3R0dNn6kpISTp48WdYslpqayptvvkleXh4Oh4N77rmH\n9957j7Fjx9K4cWMD34EwO28Xkgygonkx5wCbgD/p/2YDCwFbBds+CJwFgvT9fQ/s9kSwVpCfn290\nCD7D33LRsGFDQkJCCAkJ4YknnihbP3ToUHbv3s25c+dITU1l4sSJBkZpPH87LjzBV5uIOgBbgNBq\ntnsduAwsuG79CeBeD8QlhBBW9r+AqS/NC3FbfgFYVsE2twOlvYcBwF7gEQ/HJYQQwiTWADnAYWAt\ncI++vg2wWV/uoj9/GLADr3k5RiGEEEIIIYRwyQeOAIeAbyrZ5gPgOJAN3O+dsAyRT9W56A7sA64C\nM70XliHyqToXE1HHwxFU8+h9XovM+/KpOhePoXJxCDgARHotMu/Lp/rvC4BwoBh40gsxGSWfqnMR\nAVzSnz8E/NlbgRnhB+DOKp7/A6rTHmAg8JXHIzJOdbkIAvoDb2L9QlJdLh4AAvXlR/Hv48J9xMje\nqAtVrKq6XAA0BHYBacBTHo/IONXlIgLYWNOdeX7QHs+r6sqz0UCSvvw10Byo3VgS5lJVLs4D36KG\nlfEHVeViH+qvLVDHRTvPh2OoqnJxxW25uhuBraC6K1VfQPXXnvdCLEarLhc1vqrX7IWk9ObEb4Fn\nK3i+LeXH6zqNdb80qsuFP6lNLp7BddZqRTXJxeNALpAOvOiluIxQk++Lx4CP3La3qupy4QQGo5o9\ntwA9vRea97XWfwahruQact3zm1A3MJbaAfTFmqrLRanXsX7TVk1zMRz4DrDyoFI1zQX6c8c8HpFx\nqsvFZ6gmcIBErN20VV0umqJutwAYAeRVtTOzn5Gc1X+eB9Zz40jAPwHt3R6309dZUXW58Cc1ycV9\nqOF1RgO/eikuI9TmuNiNGu3Cd6ZyrF/V5aIf8D+o/oOngL+hjg8rqi4Xv6GGqgJ1ptqIKvpUzFxI\nrr858RHUfSjuNgKT9eVBwEXgZ69E5101yUUpXx3NoL7UJBcdgHXAJKzduVyTXNyL65goPVv/u+dD\n87qa5KIL0Fn/twZ4jlp0OJtITXLREtdxMUBfrnTEdV8btLE2WqIqKaj3sQLYDvyrvu4TVNveH1Bf\nFleoeOwuK6hJLloB+4FmgAN4CdXuedmrkXpeTXLx76jmrNK28MrmtTG7muTiKdQfW9dQx8IEL8fo\nLTXJhb+oSS7GoAppMerMxKrHhRBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgj/tory0zSHoe67\nifbAawWirtX3ptHAXC+/phBCWFIDbrzLPxg1fLi7v6Lubk70QAydqHwEAk/dLNwANbZSIw/tXwgh\nLK0TalDCJNRUze2ve/7PwFS3xw1Qk6K1AQqA29z2kwss0fezDfgn/blwXBMG/ReuQtELNWT9IdQX\neTBqXKdCfd07wDDUeFepwPf662n6/g6i5ogAiAU2oO5G/gGYAfybvs0+XINQvggcRY3gmuL2vj4C\nRlaSIyGEEFXoBJRQ+XAo6ZQfHfpBYKu+nIxrxrxOqCFFSmdWXIWadRFUYSkdQfYtVBEA+BCI0Zdv\nRRWejpQ/I4lADVPSUX88E1iqL3cDfkQVl1hUgQsA7kbNrVJaAN9DDX8DaoDS0jOPZm6vY0OdaQlR\nL8w8aKMQN+NHKp9mtSOuUVEBnkYNLY7+82m3537AVSQOoIpLIGpyqK/19StxNZ99CcwBZunbXqXi\nATS/0WMEVciW68vH9PVdUXNFZKLGj/sFNRjpJn27HH3/6PGtRBW5ErfXOOO2jRB1JoVE+Jsr1Txf\n+uXeEDWg4euoovEhqsO9dGra/3P7nRIq7tNwLxQpwCjgd9RgosNrGF9lozW7v77D7bHDLZaRwH+j\nzrL24/q834K1J20SXiaFRAiXH3FN+BOF6svogBpWvBNq6PknqfxL+BJqHofSpjP3EVO74CpIqaj5\n0f+BazjviuzG1WTWVY/le6qeCqCB288OQBYwG9fZEqj3+OMNvynETZJCIvxNVX+J7wH668sTcA21\nXWotruJw/X5KHz+DmjDrEGreh9K54ceh+k8OoTrek1HzO+xFNUf9Vd+H+37/hvqMHkF1zE9B9c1c\nv931y07UGdUyXB3176MKF6hC90VFCRBCCFE3XYDNddxHgNvybGBhHfdX325BnWmZeS4i4WPkjEQI\nl5Oopql7q9uwCiNRZx05qM7yN+shrvr0R9Tsf8VGByKEEEIIIYQQQgghhBBCCCGEEEIIIYQQQggh\nhBDC6/4fkoNSMK43/m0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc2639fad90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lZzZCCCHyVRJnNtJnI4QQotRJstEQrd+HIvGZN4mvYpNkI4QQotRJn40QQoh8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\ny5Yto3LlyuVdNSFECZJmNA0xpzZjpRQ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       "text": [
        "<matplotlib.figure.Figure at 0x7fc2639380d0>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_vdw_forces()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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YsYJp06bZTbHRczuynnMDyc/R6T0/S5DzcApAKcWlS5cYMGAAixcvxsvLy9Yh\nCSGE3bOPn+WF656a1GITY2kzvw18D60fa81HH31kwdCEEMI+SZNaIVNK8fKal9n18y52/bsLQmwd\nkRBCOA4pOHdhwf4FLN6yGH4FOsND/g/ZOqRs9NyOrOfcQPJzdHrPzxKk4OTTyZiTvLLiFfgRaA0v\ntHqBzrU62zosIYRwGNKHk78n0HROU36f/jvcgGoDq7F30F6bXU1ACCEKm/ThFBKDwcDTzk/jfMwZ\n56edWdR5kRQbIYS4S1Jw8uHSpUtMGTGFnxf/zLJ+y3jE/xFbh5QjPbcj6zk3kPwcnd7zswQ5DycP\nSileeOEF+vXrR+iTobYORwghHJb04eRh6tSpzJ49mz/++INixYpZMSwhhLBfci21gslXwbl66yrR\nkdE0b96c33//nerVqxdCaEIIYZ/koAEriYyNpNrn1Xgi9AnGfjTWoYqNntuR9ZwbSH6OTu/5WYIU\nnCxSVSphy8OIWxvHJbdLbPHcYuuQhBBCF6RJLYsv//qSwd8Nhh/A8IqBHa/voGGlhoUYnhBC2B9p\nUrOw41eP8+7ad2EF0AFGtBkhxUYIISxECk4GQzcOJXF9IlSCOs3q8L/m/7N1SHdNz+3Ies4NJD9H\np/f8LEEKTgZhZcIofqI4zu2dmdtpLm4ubrYOSQghdEP6cIxu3LhB3bp1+frrrwloEEBd37o2CE0I\nIeyTnIdTMGYLzsCBA0lOTmbWrFk2CEkIIeybHDRgIRs3bmT9+vVMnjzZ1qHcMz23I+s5N5D8HJ3e\n87OEIn0ttej4aFySXBgwYACzZ8+mTJkytg5JCCF0q8g2qSWlJNFwZkOi5kfR8r6WLJyz0NZxCSGE\n3ZImtXvw0W8fsffXvVw6dIkNQRuIT4q3dUhCCKFrRbLg/BP1D+M2jIPVQCcY2WqkbgZU03M7sp5z\nA8nP0ek9P0sokgWn3/J+pK5NhVrweLPHGdxwsK1DEkII3SuSfTh0AzZDiddLcOCNA1TzrmbrmIQQ\nwq5JH04BeWzxwLWzK5889YkUGyGEKCT2VHC8gU3AcWAj4JnDcm2Bo8C/wLsZHn8U2AnsAXYBj+S0\nokHPD+LYxGO8/MjLFgjbvui5HVnPuYHk5+j0np8l2FOT2iTgivHvu4AXMDzLMs7AMaAVcB6tsPQE\njgARwARgA9AOeAd4wsx6VEJCAsWLF7d8BnZgy5YtXLhwgZMnT5KammrrcCzq4sWLVKhQwdZhWI3k\n59gcOT/aTl/bAAAe2ElEQVSDwYCnpyctW7akdu3aOS7DPdYMezrx82mgufH+XLQCkrXgPAqcACKN\n04uBjmgF5wKQduamJ1pBMkuvxSaNi4sLI0aMwNXV1dahCCEcQEpKClFRUSxZsgQgx6Jzr+ypSc0X\niDbejzZOZ+UPnM0wfc74GGjF6TPgDPAJMMI6Ydq3PXv28OSTT0qxEULkm7OzMwEBAXTv3p0tW6w3\nynFhF5xNwAEzt6ezLKeMt6xyHqoTZgFvAJWBN4HZ9xqsIzpx4oRuL9ETGRlp6xCsSvJzbHrIz8/P\nj9jYWKu9fmE3qbXOZV40UAG4CFQELplZ5jwQkGE6AG0vB7TmtlbG+8uAmTmtKCwsjMDAQAA8PT2p\nV68eISEhQHrHn6NOX7t2jTNnzpjyS/snkGmZlmmZzmv67NmzXLhwAdC+W8LDwzPN15O0gwVAax77\n2MwyLsB/QCBQDNgL1DTO2016H1BLtAMKzFF6Nnr0aFuHUCBz5sxRjz/+eI7z27Vrp+bNm2eaHjly\npPLx8VEVK1a0yPoNBoP677//zM67ePGiatq0qfLw8FBvv/22RdZXEPYSR1GU9fOXmypVqqjNmzdb\nOSLryek7hNxbmPLFng4a+BhYCryAdlBAN+PjfsAMoD2QDLyGdiSaM1oz2hHjci8B3wBuQIJxWtiR\n7du3884773D48GGcnZ2pWbMmU6ZMoUGDBnk+d+3atab7Z86cYfLkyZw9e5ayZcsSHh7OrFmz2LZt\nm1Xinj59OuXLl+f69etWeX1Hi8PeRERE0KdPH86ePZv3wvkwZswY/vvvP+bPn296LOPnLy8GgyHt\niC6RhT0VnGukN4llFIVWbNKsM96y+htoaIW4HMrFixdtHYJZ169fp0OHDnz33Xd069aN27dvs23b\nNtzc8j+Md2RkJIGBgZw5c4ayZctStmxZK0ac7vTp09SsWTPvBc1ITk7GxSV//2Zp+RU0jpSUFJyd\nne82xEKTlt/dvCeOJK/tJ4qmwt0/LWQDBw60dQhm7dq1S3l6euY4P61J7e2331ZeXl6qatWqat26\ndab5zZs3Vx9//LHavHmzKlGihHJyclLu7u6qe/fuqnjx4srZ2Vm5u7srLy8vpZRSiYmJaujQoapy\n5crK19dXDRo0SCUkJJheb9KkSapixYrK399fzZo1K8cmtX79+ilXV1dVrFgx5e7urrZs2aJu376t\nBg8erPz8/JSfn58aMmSIun37tlJKqa1btyp/f381ceJEVaFCBdW3b1+VkpKiPvroI1WtWjXl4eGh\nHn74YXX27FmllFJHjhxRrVq1Ut7e3iooKEgtXbrU7PuTMQ4PDw+1efNmNXr0aNW5c2fVu3dvVbp0\naTVr1ix1/vx5FRoaqry9vVVwcLCaMWOG6TVGjx6tunTponr37q08PDxUnTp11PHjx9X48eNV+fLl\nVeXKldXGjRtz3EaHDx9WzZs3V56enuqBBx5QK1euzLR9Zs6cmW17pjEYDOrDDz9UwcHBKigoKNtr\nnzp1ShkMBjV9+nTl5+enKlasqD799FPT/MTERLPveXx8vCpevLjp8+Dh4aEuXLigUlNT1YQJE1S1\natVU2bJlVbdu3dS1a9cyrWvu3LmqcuXKysfHR3300UdKKaXWrVunihUrplxdXZW7u7uqV69etvxO\nnDihnnjiCVW2bFnl4+OjnnvuORUbG6tOnTqllFIqMDBQbdmyJcf30d5Zs0mtKCrcrVfI7LUP5/r1\n66ps2bKqX79+at26daZ//jRz5sxRrq6uaubMmSo1NVVNmzZN+fn5meaHhISoWbNmKaWUioiIUJUq\nVTLNCw8Pz9b/M2TIENWxY0cVExOjbty4oUJDQ9WIESOUUtqXiq+vrzp06JC6efOm6tmzZ659OGFh\nYer99983Tb///vuqcePG6vLly+ry5cuqSZMmpvlbt25VLi4uavjw4SopKUklJCSoSZMmmb7clVJq\n//796urVqyo+Pl5VqlRJhYeHq5SUFLVnzx7l4+OjDh8+nK84Ro8erVxdXdWKFSuUUkolJCSopk2b\nqldffVXdvn1b7d27V5UrV0798ssvpuWLFy+uNm7cqJKTk1Xfvn1VlSpV1Pjx41VycrKaMWOGqlq1\nqtl1JyUlqWrVqqkJEyaoO3fuqF9++UV5eHiYcsq4fdK2Z9aC06ZNGxUTE6MSExOzvX5aEejVq5e6\ndeuWOnDggCpXrpypLyS39zzr50EppaZMmaIaN26szp8/r5KSktTAgQNVz549M63rpZdeUomJiWrf\nvn3Kzc1NHT16VCml1JgxY1SfPn0yvV7G/E6cOKE2b96skpKS1OXLl1WzZs3UkCFDTMtKwREZFe7W\nK2R5FZzRW0crxpDtNnqr+eeZWz6nZfNy5MgRFRYWpipVqqRcXFzU008/raKjo5VS2hdUcHCwadmb\nN28qg8Fgmp/xH37r1q2ZvmCyfrmlpqaqUqVKZSogf/zxh+nLtH///qbio5RSx48fz7PgjBo1yjRd\nrVq1THtfGzZsUIGBgabYihUrZtrjUUqp+++/P9PeQJrFixerpk2bZnrspZdeUh988EG+4hg9erRq\n3ry5afrMmTPK2dlZxcfHmx4bMWKECgsLMy3fpk0b07yVK1cqd3d3lZqaqpTSfhQYDAYVFxeXbd2/\n/fabqlChQqbHevbsqcaMGaOUyl/B2bp1q9m8lEovAseOHTM99s4776gXXnhBKaVUUFBQru951oJT\ns2bNTF/6UVFRytXVVaWkpJjWdf78edP8Rx99VC1ZssT0PvXu3TvT62XNL6Off/5Z1a9f3zQtBSdn\n+mtILeLstQ8HoEaNGsyZMweAY8eO0bt3b4YMGcLChdpoqxkvC1KyZEkA4uPjKV++PABXrlzJ13ou\nX77MrVu3ePjhh02PKaVMl/q5cOECjzySfqm9ypUr31UeUVFRVKlSJdPzo6KiTNPlypWjWLFipumz\nZ89SrVr2i8SePn2av/76Cy8vLwBSU1NJTU2lb9+++Y6lUqVKmeLy9vamVKlSmWL7+++/TdNp7yVA\niRIl8PHxMXVwlyhRAtDe89KlS2fLOSAgINNjVapUyZR3XvLTkZ5xHZUrV+bgwYOAts1ye8+zioyM\n5JlnnsHJKf1UQxcXF6Kjo03TWT9v8fH5G4QxOjqawYMHs337dm7cuEFqaire3t7Sh5MP9nSlAVGE\n3H///fTr18/0hXIvsn6R+fj4UKJECQ4fPkxMTAwxMTHExsaaju6qWLEiZ86cMS2f8X5++Pn5mc5f\nSHu+n59fjvEEBARw4sSJbK9TuXJlmjdvbopx37593Lhxg2+++SZfcWQ9GsrPz49r165l+uI8c+ZM\npqJUUH5+fpw9exbth67m9OnT+PtrF/ooVaoUN2/eNM0z98MnPwUn63ZJe19ze8/NvW7lypVZv369\n6b2NiYnh1q1bVKxYMc8Y8orzvffew9nZmYMHDxIXF8f8+fN1d91Ca5GCozN5XTxwTMgY1GiV7TYm\nZEy+l89p2dwcO3aMyZMnc/68dom7s2fPsmjRIho3bpzv1/Dx8TH7eIUKFTh37hx37twBwMnJiRdf\nfJEhQ4Zw+fJlAM6fP8/GjRsB6NatG+Hh4Rw5coRbt27xwQcf5LrejF+yAD179mTcuHFcuXKFK1eu\n8OGHH9KnT58cnz9gwADef/99Tpw4gVKK/fv3c+3aNTp06MDx48dZsGABd+7cwd/fn127dnH06NF8\nxZF1OiAggCZNmjBixAhu377N/v37mT17Nr179841v/xo1KgRJUuWZNKkSdy5c4eIiAhWr15Njx49\nAKhXrx4//fQTCQkJnDhxglmzZmV7jfwUvnHjxpGQkMChQ4cIDw+ne/fuQO7vua+vL1evXs10uPig\nQYN47733TAXs8uXLrFy5Ml+5VqhQgcjIyGzvb5r4+HhKlSpF6dKlOX/+PJ988gmgz5MjLU0KjigU\nHh4e/PXXXzRs2BB3d3caN25M3bp1+eyzzwDz5y7k9ksz47wWLVrwwAMPUKFCBVOT0cSJEwkODqZR\no0aUKVOG1q1bc/z4cQDatm3LkCFDaNGiBdWrV6dly5Z5rivj/FGjRtGgQQPq1q1L3bp1adCgAaNG\njcox7rfeeotu3brRpk0bypQpw4svvkhiYiLu7u5s3LiRxYsX4+/vT8WKFRkxYgRJSUn5isPce7Zo\n0SIiIyPx8/Pj2Wef5cMPP6RFixY5Lp/f99zV1ZVVq1axbt06ypUrx2uvvcb8+fOpXr06AG+++SbF\nihXD19eX/v3707t372yx5kfz5s0JDg6mVatWDBs2jFattDMlcnvPa9SoQc+ePQkKCsLb25uLFy8y\nePBgnn76adq0aUPp0qVp3LgxO3fuzFc8Xbt2BaBs2bJmzxEbPXo0u3fvpkyZMoSGhtK5c2c57yaf\niuK7pHL65aIHgwYN4ttvv7V1GFah9zbyopxfZGQkQUFBJCcnZ+p3cSR62X5jxoxhzJgx2R6XET+F\nEEI4DCk4OuOoA0Dlhx5+PeamqOfn6M1Set9+liCHRQshbC4wMJCUlBRbhyGsTPZwdMaez8O5VxkP\ni9Ujyc+x6T0/S5CCI4QQolBIwdEZ6cNxXJKfY9N7fpYgBUcIIUShkIKjM9KH47gkP8em9/wsQQqO\nsAvh4eE0bdo0x/lPPfUUP/30k2l61KhRlCtXLtM1zO6Fk5MTJ0+eNDsvOjqaZs2aUbp0aYYNG2aR\n9RWEvcQh7NPLL7/MuHHj8rVsSEiI2csPWZsUHJ2x5z6c7du306RJEzw9PSlbtiyPP/54pisZ52bt\n2rW89dZbQPoQ00ePHiUqKirPYnWvMg7tnHbdLGvIqw+gsOLIzeLFi02XJ/L19aVRo0ZMmzbNND8s\nLAw3Nzc8PDzw9vamZcuWHDp0CEjP79y5czz33HP4+Pjg7u5Ow4YNWbNmTab1ODk54evrm+lQ6Tt3\n7lC+fHmzVyIICwvD1dXV7B7+hg0bTIW6fPnyhISEsGrVKgCSkpIYOnQoAQEBeHh4ULVqVd58880C\nvTe5bb/IyEicnJwsdpFPc5/5adOmZbrEUm5sNQy2FBxRKNKGmB48eDAxMTGcP3+e0aNH39UQ02kc\nbYjpworD2uexfPbZZwwZMoR3332X6OhooqOj+fbbb/n9999NF041GAy8++673Lhxg6ioKCpXrkz/\n/v1Nr3Ht2jUef/xxihcvzuHDh7l69SpvvvkmvXr14scff8y0Pm9vb9atSx9Nft26dXh7e2f7orx5\n8yY//vgjtWrVYsGCBZnmLVu2jG7duhEWFsb58+e5dOkSH374IatXrwZgwoQJ7N69m127dnHjxg0i\nIiIyDWthaXq+rJYwrzDGMLIZGWJahpi2xhDTsbGxqlSpUuqnn37KcRsqlX1U0jVr1qiSJUsqpbRB\n1kaNGqXq1KmT7XkTJ05UVapUMU0bDAb10Ucfqa5du5oe69y5s/roo4+UwWDI9Ny5c+eqOnXqqAUL\nFqjatWubHk9NTVUBAQGZhqrOqkOHDmrKlCm55pSRwWBQX375pQoKClI+Pj5q2LBhpgHsTp48qcaO\nHauqVKmiypcvr/r27WsazC4gIEAZDAbl7u6u3N3d1Z9//qmUUmrWrFmqZs2aysvLSz355JPq9OnT\nmdb17bffqvvuu095enqqV199VSmlDfVt7jPfr18/0wB9165dU+3bt1flypVTXl5eqkOHDurcuXOm\n185tQDkZ8dOy8v3hckT2WnAsMcT0xIkTlVL6HWJ6zZo1djvE9Lp165SLi4tKSUkxOz9jjGlfevHx\n8ap3797qiSeeUEppBadhw4amUUIzOnnypDIYDKb3yGAwqIMHDypfX18VFxenrl27pnx9fdXBgwez\nFZwWLVqocePGqevXr6vixYurf/75RymlFXODwaAiIyNzjHfcuHGqcuXKaurUqWr//v2m4pETg8Gg\nWrRooWJiYtSZM2dU9erV1cyZM5VSWtEMDg5Wp06dUvHx8erZZ581DVUdGRmpDAZDpvdv+fLlKjg4\nWB09elSlpKSocePGqSZNmmRaV2hoqIqLi1NnzpxR5cqVU+vXr1dKmf/MZ/x8XL16Vf30008qISFB\n3bhxQ3Xt2lV16tTJtKwUnMKT6wfK0eU1xLQtyRDTmTnSENPz58/PNsR048aNlaenpypRooTatm2b\nUkr7lV28eHHl6empnJycVFBQkLp8+bLpOcHBweq7777L9voJCQnKYDCoP/74QymlfdmeOHFCDRgw\nQH333Xdq2rRp6qWXXlInTpzIVHBOnz6tnJycTENTd+zYUQ0ePFgppdT27duVwWDItC2ySklJUd98\n84167LHHlJubm/Lz81Nz587NcXmDwaA2bNhgmp46dapq2bKlUkorfNOmTTPNO3bsWLZhrTMWnLZt\n22b60k9JSVElS5ZUZ86cMa3r999/N83v1q2b+vjjj5VS2T/zSmX/fGS0Z88e056QUrYrONKHU8Sk\ndRbe660g0oaYPnv2LAcPHiQqKoohQ4aY5uc0xPTdyjjEtJeXF15eXrRr1840RPWFCxeyDWV8N6wx\nxHTabeHChZmGQc5LfoaYThv0DvI/xHRWZcuW5cqVK5k6vf/44w9iYmIoW7as6XGDwcCwYcOIiYkh\nMjISNzc35s2bZ3qOj4+P2aGhL1y4YJqfxmAw0LdvX+bOncv8+fPp27dvtj6Q+fPnU7t2bdO4PF27\ndmXhwoWkpKSY+vjSXtscJycnXnnlFbZv305cXBwjR47k+eefz3EQPMg+DHZaPuaGwU5OTs5xe54+\nfZrBgwebtn1avBm3V9b/iYyjqubm1q1bDBw4kMDAQMqUKUPz5s2Ji4uzeR+SFBydyes8HKWURW73\nqiBDTKcVjKz0MsT0nj177HaI6caNG+Pm5sby5cvzXDbt8xEQEMCXX37J2LFjuXHjBpGRkbRq1Yqf\nfvop22do6dKlVK5cmfvuuy/T402bNuXixYtcunSJxx57LNu65s2bx7///kvFihWpWLEiQ4YM4cqV\nK6xZs4YaNWoQEBDAsmXL8pWjm5sbr7zyCl5eXhw5ciTH5bJ+dtKG2fb29s72uXBxccHX1zfHYbCn\nT5+eaRjsmzdv0qhRozxjzelHX9rjn332GcePH2fnzp3ExcXx66+/Wux/915IwRGFwhJDTOdEL0NM\n37lzx26HmPb09GT06NG88sor/Pjjj9y4cYPU1FT27t2b6Vd31phatWpFcHAwU6dOBbSRQePi4njh\nhReIjo4mMTGRRYsWMX78+BwP9V61apXZ4aF37NjByZMn2bVrF/v27WPfvn0cPHiQXr16mfaqJk+e\nzNixYwkPD+f69eukpqayfft2Bg4cCMCUKVP49ddfSUhIIDk5mblz5xIfH0/9+vVzfC8+/fRTYmNj\nOXv2LF9++aVpGOzQ0FA+//xzIiMjiY+P57333qNHjx44OTlRrlw5nJyc+O+//0yvM2jQIMaPH8/h\nw4cBiIuL44cffshxvRkLhq+vb6bPfNb58fHxlChRgjJlynDt2jWzn3FbFB8pODpjr+fhWGKI6azN\nLWn0MsR048aN7XaIaYBhw4YxefJkJk2aRIUKFahQoQKDBg1i0qRJph8O5tYxbNgwvvzyS/z9/fH2\n9mb79u0kJiZSq1YtfHx8mDJlCgsWLDAN7Zw1jlq1amU6HDxt3rx58+jUqRMPPPAA5cuXp3z58vj6\n+jJ48GDWrFlDbGwsnTt3ZsmSJcyePRt/f38qVKjA//73Pzp16gRAqVKlGDp0KBUrVqRcuXJMmzaN\nH3/8Mddzajp27MjDDz9M/fr16dChA88//zwA77zzDn369KFZs2YEBQVRsmRJvvrqK0BrDhs5ciSP\nPfYYXl5e7Ny5k06dOvHuu+/So0cPypQpQ506ddiwYUOu2ybtsZYtW2b7zGecP2TIEBISEvDx8aFJ\nkya0a9furra1tTj2iEcFo2y9W2lNOQ0PK4S4d05OTpw4cYKgoCBbh2I1MsS0yDe5lprjkvwcm97z\nswQpOEIIkU+OPgy2rckQ0zpjr304lqD38UYkP/uX2+WD9JCftckejhBCiEIhBUdnLl68aLEr0tob\nvbeRS36OTQ/5paSkWLXZUAqOzri5uREXF2frMIQQDigqKgpPT0+rvX5R7AHT9WHRW7Zs4fLly3Tu\n3BlXV1dbhyOEcAApKSlERUWxZMkS2rZtS+3atbMtY4nDoqXg6ExycjKLFy/m5MmTum1aE0JYlsFg\nwNPTk5YtW5otNmnLoKOC4w0sAaoAkUA3INbMcrOB9sAloE4Bnq/rghMREUFISIitw7AKPecGkp+j\n03t+ejvxcziwCagObDFOmzMHaHsPz9e1vXv32joEq9FzbiD5OTq952cJ9lRwngbmGu/PBTrlsNw2\nIOYenq9rsbHmdur0Qc+5geTn6PSenyXYU8HxBdIGjog2Thfm84UQQlhRYV9pYBNg7lT4kVmm73V0\nuSI7HKoezgXIiZ5zA8nP0ek9P705SnoxqmiczkkgcKCAzz9BekGSm9zkJje55e+WfRTBu2RP11Jb\nCfQDJhr/5j20YMGeH1zQAIUQQuiDN7AZOA5sBNJOd/UD1mRYbhEQBdwGzgL983i+EEIIIYQQQjie\n2WhHp2Xs2/FGO1Ahr72etmh9Pv8C71oxxoK6l9wigf3AHmCn9UK8J+by6wocAlKAh3J5rr1vO7i3\n/CJxzO33CXAE2Af8BJTJ4bmOuv3ym18k9r39zOU2Fi2vvWjnNAbk8FxH2HZW0xSoT+Y3bhLwjvH+\nu8DHZp7njNYZFgi4or3JNc0sZ0sFzQ3gFFpxsmfm8quBdhLvVnL+QnaEbQcFzw8cd/u1Jv20i49x\n3P89KHh+YP/bz1xuHhnuvw7MNPO8Am07ezoP516ZOyE0PyeDPor2xkUCd4DFQEfrhFhgBc0tjT1d\nwsgcc/kdRdt7y40jbDsoeH5pHHH7bQLSLub3F1DJzPMcefvlJ7809rz9zOV2I8N9d+CKmecVaNvp\nqeCYk5+TQf3RDj5Ic874mL3L74muCu1gir+BFwshrsLkqNvubuhh+z0PrDXzuF62X075geNuv4+A\nM2hH/JrbeyvQtrOnw6KtLe1YcnOPO7qccgN4DLgAlEP7VXYU7VeNHuhh2+XF0bffSCAJWGhmnh62\nX275geNuv5HG23Dgc9KPBk5ToG2n9z2caDKfDHrJzDLnydwpFoBWre1dfnID7cMOcBn4GW1XWC8c\nddvdDUfefmHAU8BzOcx39O0XRu75gWNvP9AK6SNmHi/QttN7wUk7GRRyPhn0b+A+tM6vYkB34/Ps\nXX5yK0l6B2ApoA3Zr9DgCHJqA3fUbZdVTvk58vZrCwxDa9dPzGEZR95++cnPUbfffRnud0Q7wi4r\nR952FpF2QmgS6SeE5vdk0nbAMbROsBGFFO/dKGhuQWhHj+wFDmKfuUH2/J5HOwjiLJAAXATWGZd1\ntG0HBc/Pkbffv8BptC+rPcBU47J62X75yc8Rtp+53JahFca9wI9AeeOyjrjthBBCCCGEEEIIIYQQ\nQgghhBBCCCGEEEIIIYQQQghR+JYA1TJM10O7eOOTVlhXGeBlK7xubp4G3i/kdQohRJFmIPuZ/8HA\n6iyPTUQ7szrcCjEEkvMZ6da6/qEB7UQ/Vyu9vhBCCLQv+GNowzwcJPuAU6OAlzJMG9DONPdDu6Ku\nW4bXOQJMN77OBqC4cd4jpA/E9QnpBeUBtMvb70H7wg9Gu9T7LeNjk4DmaBd8XIF28Uc3YI7x9XYD\nIcbXCkO7xNFGtHFYXgPeNi6zA/AyLvcG2uBv+9DONE8zDWifw3skhBDCAgLRRt7M6cKK68g8SNpj\nwHrj/XnAsxle5w5Q1zi9hPQLOx4EGhrvT0ArFgBfAb2M913QClQVMu/hhADxxscBhpI+KNb9aJdY\ncUMrOP+iXbfLB4gjvVBOBgYb758nfU+mdIb19EfbcxPCKvR+8U4h8us0OQ8BXIX0q/4C9AR+MN7/\nwTid5hTpxeQftCJUBm0gq7+Mjy8kvdnuD+A9tNFbA9EuBGnuYp47jTGCVvAWGO8fMz5eHe2S8VuB\nm2iDZsUCq4zLHTC+Psb4FqIVw5QM64jKsIwQFicFRwjNzTzmpxUBZ6AzMBqtuHyFduBAKeP82xme\nk4L5PpeMBWUREIp2Ec+1wBP5jC+nK0xnXH9qhunUDLG0B75B22vbRfr3gBP6GKNG2CkpOELk7TTa\nmEMALdH6WioDVdH2CH5Ca1bL6cs6Dm3Y3rQmux4Z5gWRXrhWAHWA62QeVz6rbaQ31VU3xnKU3Icy\nNmT4WxmIQBtcK23vC7QcT2d7phAWIgVHCE1uv+y3Aw2M93ugDaSV0Y+kF5Gsr5M2/QIwA+1AgJJo\nRQigG1r/zh60AwjmAdeA39GawSaSfUTXqWj/u/vRDjDoh9Z3lHW5rPcV2h7afNIPOPgCrcCBVhB/\nM/cGCCGEKBxBZB4HpCBKZbifNmyvPXFC23MrSsPOi0ImezhC5O0kWpNYtbwWzEV7tL2YA2id/uMs\nEJcldUAbeCvZ1oEIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEHbq/zofoUrjMPF3AAAAAElF\nTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fc263b45ad0>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And all was good in the world."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}