Difference in purchase cost between gas pump or heat pump

It won’t work quite like that. There are plenty of days where there isn’t even enough photovoltaic power to cover the base load.
I installed a Geisha (F model) and also experimented with timers, lowering it at night and raising it a bit during the day. Often, it wouldn’t heat at all at night and the house cooled down a bit.
None of that really convinced me. The only thing I do is have the hot water generated only between 11 a.m. and 5 p.m. That’s usually enough for us. But it depends on showering habits. And our house/photovoltaics face southeast, so there is more sun in the morning than in the afternoon.
There are certainly other photovoltaic-heat pump combinations that communicate better with each other and thus achieve a higher self-consumption rate.
Still, I am very satisfied with the Geisha, especially for the price of 2,500€. But you have to do a lot yourself or put effort into it. Or have a capable and willing heating installer who supports you with such an installation.

How much electricity do you have to purchase annually with how many m² of living space?
And what is your annual performance factor?

You mean "Smart controlled you ideally only heat the water at midday when the sun is shining on your photovoltaic system, with which you can then heat as needed at night." because especially at midday the house should not be heated in addition to the solar energy.

By smart I always imagine being able to control everything via the internet on a smartphone or something. Or raise/lower the blinds via smartphone. I usually don’t need that and have no plans to install it. But to use my own photovoltaic electricity through the heat pump, my house doesn’t have to be smart, right? That should be standard, even without smart?

By smart I meant rather intelligently controlled and not necessarily with smart home.
It goes in the direction of what described.

I do hope that from now on only those heat pumps are installed that are intelligently controlled ("smart"). Anything else would be a waste of money (at least currently with the feed-in tariff, etc.)!?

How much electricity do you have to purchase per year for how many m² of living space?
And what is your annual performance factor?

Unfortunately, I cannot provide complete values yet, as we only moved in at the end of February 2019.
KfW40 house with controlled residential ventilation
143 m² living space, plus approx. 70 m² basement. Everything except the storage room in the basement is heated with underfloor heating. In the main bathroom, there is additionally wall heating.
Standard outside temperature -16°C (postal code 88***)

01.01.2019 to 01.12.19:
Produced heat 7216 kWh
Electrical work 1867 kWh
Annual performance factor 3.86

The annual performance factor could presumably be higher, but due to the very well insulated house, the hot water preparation has a higher impact on the annual performance factor.

At the beginning, the heating element for defrosting was still activated, which used quite a bit of electricity. However, it has been off for a long time.

Photovoltaics: 9.74 kWp
Current values 2019 to today:
Photovoltaic production: 11070 kWh
Direct consumption: 1490 kWh
Fed in: 9580 kWh
Electricity purchased from the grid: 2386 kWh (total household electricity)

We have also installed the Geisha (the new one) since the beginning of 2018:
- clean a filter once a year, otherwise no maintenance
- if it breaks, it is simply bought new, 2500€
- heating element is disconnected (-12 Norm AT, so relatively mild)
- I have not done anything myself, except programming and commissioning

I have refrained from smart control for photovoltaics; in my opinion, it doesn't help. I ensure that it ALWAYS runs during the day in the heating period - for me from 9 am to at least 8 pm. Because then it is warmer and the sun could shine. At night, when it is coldest, it turns off for 6 to 11 hours depending on the outside temperature. If it is very cold, it runs continuously. I shift the heating curve accordingly by timer.

Throughout the year, maybe 30% of the heating and hot water is contributed by photovoltaics.

You can also simply run it 24/7.

We have also installed the Geisha (the new one) since early 2008:
- clean the filter once a year, otherwise no maintenance
- if defective, it is simply bought new, 2500€
- heating element is disconnected (-12 Norm AT, so relatively mild)
- I have done nothing myself except the programming and commissioning

I have refrained from smart control for photovoltaics, in my opinion it brings nothing. I make sure that it ALWAYS runs during the day in the heating period – for me from 9 am to at least 8 pm. Because then it is warmer and the sun could shine. At night, when it is coldest, it turns off depending on the outside temperature for 6 to 11 hours. If it is very cold, it runs continuously. I shift the heating curve accordingly by timer.

Over the year, photovoltaic contributes maybe 30% to heating and hot water.

You can also simply let it run 24/7.

But you procured it yourself? It would probably be more expensive through a heating installer.

I bought them myself, but only because I couldn't find anyone who wanted to sell and install them for me. I would have paid more.

The heating engineer did everything else: underfloor heating and hydraulics – so everything in the house – and kindly also the 2 water-carrying pipes up to the heat pump outside. But of course, it's excluded from warranty etc.